TOHOKU UNIVERSITY Researchers

Toshiyuki Sugawa

Section

Graduate School of Information Sciences

Job title

Professor

Degree

博士（理学）（京都大学）
理学修士（京都大学）

researchmap

https://researchmap.jp/read0140757

J-GLOBAL ID

200901008960739700

Research History 3

2008/04 - Present

Tohoku University　Graduate School of Information Sciences　Professor
2002/04 - 2008/03

Hiroshima University　Graduate School of Science　Associtate Professor
1991/04 - 2002/03

Kyoto University　Graduate School of Science Division of Mathematics　Assistant

Education 3

Kyoto University　Graduate School of Science　Mathematics

1990/04 - 1991/03
Kyoto University　Graduate School of Science　Mathematics

1988/04 - 1990/03
Kyoto University　Faculty of Science

1984/04 - 1988/03

Professional Memberships 2

American Mathematical Society
Mathematical Society of Japan

Research Interests 1

Complex Analysis

Research Areas 1

Natural sciences / Basic analysis /

Papers 81

Nonlinear Resolvents and Decreasing Loewner Chains

Ikkei Hotta, Sebastian Schleißinger, Toshiyuki Sugawa

Journal of Geometric Analysis　34　(4)　2024/04

DOI： 10.1007/s12220-023-01544-y 　

ISSN：1050-6926

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In this article,we prove that nonlinear resolvents of infinitesimal generators on bounded and convex subdomains of Cn are decreasing Loewner chains. Furthermore, we consider the problem of the existence of nonlinear resolvents on unbounded convex domains in C. In the case of the upper half-plane, we obtain a complete solution by using that nonlinear resolvents of certain generators correspond to semigroups of probability measures with respect to free convolution.
Successive coefficients for functions in the spirallike family

Vibhuti Arora, Saminathan Ponnusamy, Swadesh Kumar Sahoo, Toshiyuki Sugawa

Bulletin des Sciences Mathematiques　188　2023/11

DOI： 10.1016/j.bulsci.2023.103323 　

ISSN：0007-4497

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The aim of this paper is to compute the bounds of successive coefficients ||an+1|−|an|| for the family of univalent functions that are spirallike functions of non-negative order. This result not only improves a recent result but gives an alternate simple proof for the case of univalent spirallike functions.
Moduli of quadrilaterals and quasiconformal reflection

Semen Nasyrov, Toshiyuki Sugawa, Matti Vuorinen

Journal of Mathematical Analysis and Applications　524　(2)　2023/08/15

DOI： 10.1016/j.jmaa.2023.127092 　

ISSN：0022-247X

eISSN：1096-0813

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We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. Then we study the special case of an isosceles trapezoidal polygon L and obtain two-sided estimates for the coefficient of quasiconformal reflection over L in terms of special functions and geometric parameters of L.
Conformally invariant complete metrics

Toshiyuki Sugawa, Matti Vuorinen, Tanran Zhang

MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY　174　(2)　273-300　2022/05
Publisher: CAMBRIDGE UNIV PRESS
DOI： 10.1017/SO30500412200024X 　

ISSN：0305-0041

eISSN：1469-8064

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For a domain G in the one-point compactification (R) over bar (n) = R-n boolean OR{infinity} of R-n , n >= 2, we charac- terise the completeness of the modulus metric mu(G) in terms of a potential-theoretic thickness condition of partial derivative G, Martio's M-condition [35]. Next, we prove that partial derivative G is uniformly perfect if and only if mu(G) admits a minorant in terms of a Mobius invariant metric. Several applications to quasiconformal maps are given.
Geometric deduction of the solutions to modular equations

Md Shafiul Alam, Toshiyuki Sugawa

Ramanujan Journal　59　(2)　459-477　2022

DOI： 10.1007/s11139-022-00604-1 　

ISSN：1382-4090

eISSN：1572-9303

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In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series and hypergeometric functions. We offer a geometric approach to the proof of those formulae. We emphasize that our proofs are geometric and independent of such identities.
Harmonic spirallike functions and harmonic strongly starlike functions

Xiu Shuang Ma, Saminathan Ponnusamy, Toshiyuki Sugawa

Monatshefte fur Mathematik　199　(2)　363-375　2022

DOI： 10.1007/s00605-022-01708-y 　

ISSN：0026-9255

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In this note, we define two subclasses of normalized harmonic univalent functions of the unit disk, spirallike functions and strongly starlike functions, which preserve a hereditary property and have nice analytic and geometric characterizations. We also investigate the uniform boundedness and quasiconformal extendability of strongly starlike functions. Some coefficient conditions can be given for strong starlikeness and spirallikeness. We consider a special form of harmonic functions as an application.
Intrinsic Geometry and Boundary Structure of Plane Domains

O. Rainio, T. Sugawa, M. Vuorinen

Siberian Mathematical Journal　62　(4)　691-706　2021/07

DOI： 10.1134/S0037446621040121 　

ISSN：0037-4466

eISSN：1573-9260

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Given a nonempty compact set E in a proper subdomain Ω of the complex plane,we denote the diameter of E andthe distance from E to the boundary of Ω by d(E) and d(E,∂Ω), respectively.The quantity d(E)/d(E,∂Ω) is invariant under similarities andplays an important role in geometric function theory.In case Ω has the hyperbolic distance hΩ(z,w), we consider the infimum k(Ω) ofthe quantity hΩ(E)/log(1+d(E)/d(E,∂Ω)) over compact subsetsEof Ω with at least two points, where hΩ(E) stands forthe hyperbolic diameter of E.Let the upper half-plane be H. We show that k(Ω) is positive if and only if theboundaryof Ω is uniformly perfect and k(Ω) ≤ k(H) for all Ω, with equality holding precisely when Ω is convex.
On the ninth coefficient of the inverse of a convex function

Toshiyuki Sugawa

Mathematics　9　(7)　2021/04

DOI： 10.3390/math9070706 　

eISSN：2227-7390

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We consider the inverse function z = g(w) = w + b w +…of a normalized convex univalent function w = f (z) = z + a z +…on the unit disk in the complex plane. So far, it is known that {divides}b {divides} ≥ 1 for n = 2, 3,…, 8. On the other hand, the inequality {divides}b {divides} ≥ 1 is not valid for n = 10. It is conjectured that {divides}b {divides} ≥ 1. The present paper offers the estimate {divides}b {divides} < 1.617. 2 2 n n 9 9 2 2
Sharp inequalities for logarithmic coefficients and their applications

Saminathan Ponnusamy, Toshiyuki Sugawa

Bulletin des Sciences Mathematiques　166　2021/02

DOI： 10.1016/j.bulsci.2020.102931 　

ISSN：0007-4497

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I. M. Milin proposed, in his 1971 paper, a system of inequalities for the logarithmic coefficients of normalized univalent functions on the unit disk of the complex plane. This is known as the Milin conjecture and implies the Robertson conjecture which in turn implies the Bieberbach conjecture. In 1984, Louis de Branges settled the long-standing Bieberbach conjecture by showing the Milin conjecture. Recently, O. Roth proved an interesting sharp inequality for the logarithmic coefficients based on the proof by de Branges. In this paper, following Roth's ideas, we will show more general sharp inequalities with convex sequences as weight functions. By specializing the sequence, we can obtain an abundant number of sharp inequalities on logarithmic coefficients, some of which are provided in Appendix. We also consider the inequality with the help of de Branges system of linear ODE for non-convex sequences where the proof is partly assisted by computer. Also, we apply some of those inequalities to improve previously known results.
On the Fourth Coefficient of the Inverse of a Starlike Function of Positive Order

SUGAWA Toshiyuki, WANG Li-Mei

Interdisciplinary Information Sciences　27　(1)　93-106　2021
Publisher: Graduate School of Information Sciences, Tohoku University
DOI： 10.4036/iis.2021.R.01 　

ISSN：1340-9050

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We consider the inverse function z = g(w) of a (normalized) starlike function w = f(z) of order α on the unit disk of the complex plane with 0 < α < 1. Krzyż, Libera and Złotkiewicz obtained sharp estimates of the second and the third coefficients of g(w) in their paper (1979). Prokhorov and Szynal gave sharp estimates of the fourth coefficient of g(w) as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of g(w) together with explicit forms of the extremal functions.
Teichmüller’s Theorem in Higher Dimensions and Its Applications

Anatoly Golberg, Toshiyuki Sugawa, Matti Vuorinen

Computational Methods and Function Theory　20　(3-4)　539-558　2020/11

DOI： 10.1007/s40315-020-00340-x 　

ISSN：1617-9447

eISSN：2195-3724

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For a given ring (domain) in R¯ , we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all n≥ 3 , the standard definition of uniformly perfect sets in terms of the Euclidean metric is equivalent to the boundedness of the moduli of the separating rings. We also establish separation theorems for a “half” of a ring. As applications of those results, we will prove boundary Hölder continuity of quasiconformal mappings of the ball or the half space in R . n n
On the hyperbolic distance of n-times punctured spheres

Toshiyuki Sugawa, Matti Vuorinen, Tanran Zhang

Journal d'Analyse Mathematique　141　(2)　663-687　2020/09

DOI： 10.1007/s11854-020-0112-9 　

ISSN：0021-7670

eISSN：1565-8538

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The shortest closed geodesic in a hyperbolic surface X is called a systole of X. When X is an n-times punctured sphere ℂ^ \ A where A⊂ ℂ^ is a finite set of cardinality n ≥ 4, we define a quantity Q(A) in terms of cross ratios of quadruples in A so that Q(A) is quantitatively comparable with the systole length of X. We next propose a method to construct a distance function d onapunctured sphere X which is Lipschitz equivalent to the hyperbolic distance h on X. In particular, when the construction is based on a modified quasihyperbolic metric, d is Lipschitz equivalent to h with a Lipschitz constant depending only on Q(A). X X X X
Aharonov invariants revisited

Toshiyuki Sugawa

Analysis and Mathematical Physics　10　(3)　2020/09/01

DOI： 10.1007/s13324-020-00374-3 　

ISSN：1664-2368

eISSN：1664-235X

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In this note, we reformulate Aharonov’s univalence criterion in terms of projective Schwarzian derivatives of higher order. We also mention some relations between quasiconformal extendibility and the Aharonov criterion.
Geometric properties of the nonlinear resolvent of holomorphic generators

Mark Elin, David Shoikhet, Toshiyuki Sugawa

Journal of Mathematical Analysis and Applications　483　(2)　2020/03/15

DOI： 10.1016/j.jmaa.2019.123614 　

ISSN：0022-247X

eISSN：1096-0813

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Let f be the infinitesimal generator of a one-parameter semigroup of holomorphic self-mappings of the open unit disk Δ. Our main purpose is to study properties of the family R of non-linear resolvents (I+rf) :Δ→Δ,r≥0, in the spirit of classical geometric function theory. To make a connection with this theory, we mostly consider the case where f(0)=0 and f (0) is a positive real number. We found, in particular, that R forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of R satisfies the Noshiro-Warschawski condition. This, in turn, implies that each element of R is also the infinitesimal generator of a one-parameter semigroup on Δ. We consider also quasiconformal extensions of elements of R. Finally we study the existence of repelling fixed points of this family. −1 ′
Schur Parameters and the Carathéodory Class

Ming Li, Toshiyuki Sugawa

Results in Mathematics　74　(4)　2019/12/01

DOI： 10.1007/s00025-019-1107-7 　

ISSN：1422-6383

eISSN：1420-9012

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The Schur (resp. Carathéodory) class consists of all the analytic functions f on the unit disk with | f| ≤ 1 (resp. Ref>0 and f(0) = 1). The Schur parameters γ, γ , ⋯ (| γ | ≤ 1) are known to parameterize the coefficients of functions in the Schur class. By employing a recursive formula for it, we describe the n-th coefficient of a Carathéodory function in terms of n independent variables γ , ⋯ , γ . The mapping properties of those correspondences are also studied. 1 j 1 n
Hornich Operations on Functions of Bounded Boundary Rotations and Order α

Saminathan Ponnusamy, Swadesh Kumar Sahoo, Toshiyuki Sugawa

Computational Methods and Function Theory　19　(3)　455-472　2019/09/01

DOI： 10.1007/s40315-019-00276-x 　

ISSN：1617-9447

eISSN：2195-3724

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In this paper, we consider the class of functions of bounded boundary rotation and some generalizations and refinements of the class. We will see that several results can be understood more naturally by introducing the Hornich operations to these classes. In this context, we are particularly interested in linear integral transformations involving the Gauss hypergeometric function. We apply these ideas to the study of actions on those classes and univalence properties of our transformations.
Characterizations for Inner Functions in Certain Function Spaces

Atte Reijonen, Toshiyuki Sugawa

Complex Analysis and Operator Theory　13　(4)　1853-1871　2019/06/01

DOI： 10.1007/s11785-018-0863-9 　

ISSN：1661-8254

eISSN：1661-8262

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For 12<p<∞, 0 < q< ∞ and a certain two-sided doubling weight ω, we characterize those inner functions Θ for which ‖Θ′‖Aωp,qq=∫01(∫02π|Θ′(reiθ)|pdθ)q/pω(r)dr<∞.Then we show a modified version of this result for p⩾ q. Moreover, two additional characterizations for inner functions whose derivative belongs to the Bergman space Aωp,p are given.
A construction of trivial beltrami coefficients

Toshiyuki Sugawa

Proceedings of the American Mathematical Society　147　(2)　629-635　2019/02

DOI： 10.1090/proc/13965 　

ISSN：0002-9939

eISSN：1088-6826

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A measurable function μ on the unit disk D of the complex plane with ‖μ‖ < 1 is sometimes called a Beltrami coefficient. We say that μ is trivial if it is the complex dilatation (Formula Presented) of a quasiconformal automorphism f of D satisfying the trivial boundary condition f (z) = z, |z| = 1. Since it is not easy to solve the Beltrami equation explicitly, to detect triviality of a given Beltrami coefficient is a hard problem, in general. In the present article, we offer a sufficient condition for a Beltrami coefficient to be trivial. Our proof is based on Betker’s theorem on Löwner chains. ∞
An Extremal Problem for Univalent Functions

SUGAWA Toshiyuki, WANG Li-Mei

Interdisciplinary Information Sciences　25　(1)　45-52　2019
Publisher: Graduate School of Information Sciences, Tohoku University
DOI： 10.4036/iis.2019.A.04 　

ISSN：1340-9050

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For a real constant b, we give sharp estimates of log |f(z)/z|+b arg[f(z)/z] for subclasses of normalized univalent functions f on the unit disk.
Hereditarily non uniformly perfect sets

Rich Stankewitz, Toshiyuki Sugawa, Hiroki Sumi

Discrete and Continuous Dynamical Systems - Series S　13　(2)　2391-2402　2019

DOI： 10.3934/dcdss.2019150 　

ISSN：1937-1632

eISSN：1937-1179

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We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give a detailed construction of a compact set in the plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect.
Construction of nearly hyperbolic distance on punctured spheres

Toshiyuki Sugawa, Tanran Zhang

Bulletin of Mathematical Sciences　8　(2)　307-323　2018/08/01

DOI： 10.1007/s13373-017-0111-z 　

ISSN：1664-3607

eISSN：1664-3615

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We define a distance function on the bordered punctured disk 0 < | z| ≤ 1 / e in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk 0 < | z| < 1. As an application, we will construct a distance function on an n-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute.
Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions Peer-reviewed

Rintaro Ohno, Toshiyuki Sugawa

Kyoto Journal of Mathematics　58　(2)　227-241　2018/06/01

DOI： 10.1215/21562261-2017-0015 　

ISSN：2156-2261

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In this article, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete–Szeg o problem for normalized concave functions with a pole in the unit disk.
Filtration of Semi-complete Vector Fields Revisited

Mark Elin, David Shoikhet, Toshiyuki Sugawa

Trends in Mathematics　(9781493976584)　93-102　2018

DOI： 10.1007/978-3-319-70154-7_6 　

ISSN：2297-0215

eISSN：2297-024X

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In this note we use a technique based on subordination theory in order to continue the study of filtration theory of semi-complete vector fields.
Geometric Properties of the Shifted Hypergeometric Functions Peer-reviewed

Toshiyuki Sugawa, Li Mei Wang

Complex Analysis and Operator Theory　11　(8)　1879-1893　2017/12/01
Publisher: SPRINGER BASEL AG
DOI： 10.1007/s11785-017-0674-4 　

ISSN：1661-8254

eISSN：1661-8262

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We will provide sufficient conditions for the shifted hypergeometric function z F (a, b; c; z) to be a member of a specific subclass of starlike functions in terms of the complex parameters a, b and c. For example, we study starlikeness of order α, λ-spirallikeness of order α and strong starlikeness of order α. In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk. 2 1
A Note on Successive Coefficients of Convex Functions

Ming Li, Toshiyuki Sugawa

Computational Methods and Function Theory　17　(2)　179-193　2017/06/01

DOI： 10.1007/s40315-016-0177-8 　

ISSN：1617-9447

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In this note, we investigate the supremum and the infimum of the functional | a | - | a | for functions, convex and analytic on the unit disk, of the form f(z) = z+ a z + a z + ⋯. We also consider the related problem of maximizing the functional | a - a | for convex functions f with f (0) = p for a prescribed p∈ [0 , 2]. n + 1 n 2 3 n + 1 n 2 3 ′ ′
Spirallikeness of shifted hypergeometric functions

Toshiyuki Sugawa, Li Mei Wang

Annales Academiae Scientiarum Fennicae Mathematica　42　963-977　2017

DOI： 10.5186/AASFM.2017.4257 　

ISSN：1239-629X

eISSN：1798-2383

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In the present paper, we study spirallikenss (including starlikeness) of the shifted hypergeometric function f(z) = F (a, b; c; z) with complex parameters a, b, c, where F (a, b; c; z) stands for the Gaussian hypergeometric function. First, we observe the asymptotic behaviour of F (a, b; c; z) around the point z = 1 to obtain necessary conditions for f to be λ-spirallike for a given λ with -π/2 < λ < π/2. We next give sufficient conditions for f to be λ-spirallike. As special cases, we obtain sufficient conditions of strong starlikeness and examples of spirallike, but not starlike, shifted hypergeometric functions. z2 1 2 1 2 1
ANALYTIC AND GEOMETRIC PROPERTIES OF OPEN DOOR FUNCTIONS Peer-reviewed

Ming Li, Toshiyuki Sugawa

JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY　54　(1)　267-280　2017/01
Publisher: KOREAN MATHEMATICAL SOC
DOI： 10.4134/JKMS.j150720 　

ISSN：0304-9914

eISSN：2234-3008

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In this paper, we study analytic and geometric properties of the solution q (z) to the differential equation q(z) +zq' (z) / q (z) = h(z) with the initial condition q(0) = 1 for a give ananalytic function h(z) on the unit disk vertical bar z vertical bar < 1 in the complex plane with h(0) = 1. In particular, we investigate the possible largest constant c > 0 such that the condition vertical bar Im [zf '' (z) /f' (z)]vertical bar < c on vertical bar z vertical bar < 1 implies starlikeness of an analytic function f (z) on vertical bar z vertical bar < 1 with f(0)= f '(0) - 1 = 0.
QUASICONFORMAL EXTENSION OF MEROMORPHIC FUNCTIONS WITH NONZERO POLE Peer-reviewed

B. Bhowmik, G. Satpati, T. Sugawa

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY　144　(6)　2593-2601　2016/06
Publisher: AMER MATHEMATICAL SOC
DOI： 10.1090/proc/12933 　

ISSN：0002-9939

eISSN：1088-6826

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In this note, we consider meromorphic univalent functions f(z) in the unit disc with a simple pole at z = p is an element of (0, 1) which have a k-quasiconformal extension to the extended complex plane (C) over cap, where 0 <= k < 1. We denote the class of such functions by Sigma(k)(p). We first prove an area theorem for functions in this class. Next, we derive a sufficient condition for meromorphic functions in the unit disc with a simple pole at z = p is an element of(0, 1) to belong to the class Sigma(k)(p). Finally, we give a convolution property for functions in the class Sigma(k)(p).
Notes on Convex Functions of Order alpha Peer-reviewed

Toshiyuki Sugawa, Li-Mei Wang

COMPUTATIONAL METHODS AND FUNCTION THEORY　16　(1)　79-92　2016/03
Publisher: SPRINGER HEIDELBERG
DOI： 10.1007/s40315-015-0122-2 　

ISSN：1617-9447

eISSN：2195-3724

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Marx and Strohhacker showed in 1933 that f(z) / z is subordinate to for a normalized convex function f on the unit disk In 1973, Brickman, Hallenbeck, MacGregor and Wilken further proved that f(z) / z is subordinate to if f is convex of order for and conjectured that this is true also for Here, is the standard extremal function in the class of normalized convex functions of order and We prove the conjecture and study geometric properties of convex functions of order In particular, we prove that is starlike whenever both f and g are convex of order 3 / 5.
Variability regions of close-to-convex functions Peer-reviewed

Takao Kato, Toshiyuki Sugawa, Li-Mei Wang

ANNALES POLONICI MATHEMATICI　111　(1)　89-105　2014
Publisher: POLISH ACAD SCIENCES INST MATHEMATICS
DOI： 10.4064/ap111-1-7 　

ISSN：0066-2216

eISSN：1730-6272

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M. Biernacki gave in 1936 concrete forms of the variability regions of z/f(z) and zf'(z)/f(z) of close-to-convex functions f for a fixed z with vertical bar z vertical bar < 1. The forms are, however, not necessarily convenient to determine the shape of the full variability region of zf'(z)/f(z) over all close-to-convex functions f and all points z with vertical bar z vertical bar < 1. We propose a couple of other forms of the variability regions and see that the full variability region of zf'(z)/f(z) is indeed the complex plane minus the origin. We also apply them to study the variability regions of log[z/f(z)] and log[zf(z)/f(z)].
Univalence criteria and analogues of the John constant Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

Bulletin of the Australian Mathematical Society　88　(3)　423-434　2013/12

DOI： 10.1017/S0004972712000962 　

ISSN：0004-9727 1755-1633

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Let p(z) = z f 0(z)= f (z) for a function f (z) analytic on the unit disc j z j&lt 1 in the complex plane and normalised by f (0) = 0 f 0(0) = 1. We provide lower and upper bounds for the best constants δ0 and δ1 such that the conditions e. δ0=2 &lt j p(z) j&lt e0=2 and j p(w)=p(z) j&lt e1 for j z j j w j&lt 1 respectively imply univalence of f on the unit disc. © 2012 Australian Mathematical Publishing Association Inc.
On mu-conformal homeomorphisms and boundary correspondence Peer-reviewed

Vladimir Gutlyanskii, Ken-ichi Sakan, Toshiyuki Sugawa

COMPLEX VARIABLES AND ELLIPTIC EQUATIONS　58　(7)　947-962　2013/07
Publisher: TAYLOR & FRANCIS LTD
DOI： 10.1080/17476933.2011.613116 　

ISSN：1747-6933

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We study the boundary correspondence under -homeomorphisms f of the open upper half-plane onto itself. Sufficient conditions are given for f to admit a homeomorphic extension to the closed half-plane with prescribed boundary regularity. The proofs are based on the modulus estimates for semiannuli in terms of directional dilatations of f which might be of independent interest.
Univalent Functions with Half-Integral Coefficients Peer-reviewed

Naoki Hiranuma, Toshiyuki Sugawa

COMPUTATIONAL METHODS AND FUNCTION THEORY　13　(1)　133-151　2013
Publisher: HELDERMANN VERLAG
DOI： 10.1007/s40315-013-0009-z 　

ISSN：1617-9447

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B. Friedman discovered in his 1946 paper that the set of analytic univalent functions on the unit disk in the complex plane with integral Taylor coefficients consists of nine functions. In the present paper, we prove that the similar set obtained by replacing "integral" by "half-integral" consists of another 12 functions in addition to the nine. We also observe geometric properties of the 12 functions.
On univalence of the power deformation z(f(z)/z)(c) Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

CHINESE ANNALS OF MATHEMATICS SERIES B　33　(6)　823-830　2012/11
Publisher: SHANGHAI SCIENTIFIC TECHNOLOGY LITERATURE PUBLISHING HOUSE
DOI： 10.1007/s11401-012-0750-z 　

ISSN：0252-9599

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The authors mainly concern the set U (f) of c a a", such that the power deformation is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a (2) z (2) + aEuro broken vertical bar in the unit disk. It is shown that U (f) is a compact, polynomially convex subset of the complex plane a", unless f is the identity function. In particular, the interior of U (f) is simply connected. This fact enables us to apply various versions of the lambda-lemma for the holomorphic family of injections parametrized over the interior of U (f) . The necessary or sufficient conditions for U (f) to contain 0 or 1 as an interior point are also given.
On power deformations of univalent functions Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

MONATSHEFTE FUR MATHEMATIK　167　(2)　231-240　2012/08
Publisher: SPRINGER WIEN
DOI： 10.1007/s00605-011-0344-y 　

ISSN：0026-9255

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For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f'(0) - 1 = 0 and f (z) not equal 0, 0 < |z| < 1, we consider the power deformation f (c) (z) = z(f (z)/z) (c) for a complex number c. We determine those values c for which the operator maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf'(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.
Correspondence between spirallike functions and starlike functions Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

MATHEMATISCHE NACHRICHTEN　285　(2-3)　322-331　2012/02
Publisher: WILEY-BLACKWELL
DOI： 10.1002/mana.201010020 　

ISSN：0025-584X

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Let ? be a real number with -p/2 < ? < p/2. In order to study ?-spirallike functions, it is natural to measure the angle according to ?-spirals. Thus we are led to the notion of ?-argument. This fits well the classical correspondence between ?-spirallike functions and starlike functions. Using this idea, we extend deep results of Pommerenke and Sheil-Small on starlike functions to spirallike functions. As an application, we solved a problem given by Hansen in 6.
A Remark on Löwner's Theorem

SUGAWA Toshiyuki

Interdisciplinary Information Sciences　18　(1)　19-22　2012
Publisher: The Editorial Committee of the Interdisciplinary Information Sciences
DOI： 10.4036/iis.2012.19 　

ISSN：1340-9050

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We give a refinement of Löwner's inequality with argument of the equality case. To this end, we establish a complete univalence criterion for meromorphic functions of special type. We also give applications of the refinement.
On a Multi-Point Schwarz-Pick Lemma Peer-reviewed

Kyung Hyun Cho, Seong-A Kim, Toshiyuki Sugawa

COMPUTATIONAL METHODS AND FUNCTION THEORY　12　(2)　483-499　2012
Publisher: HELDERMANN VERLAG
DOI： 10.1007/BF03321839 　

ISSN：1617-9447

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We consider the multi-point Schwarz-Pick Lemma and its associate functions due to Beardon-Minda and Baribeau-Rivard-Wegert. Basic properties of the associate functions are summarized. Then we observe that special cases of the multi-point Schwarz-Pick Lemma give the Schur's continued fraction algorithm and several inequalities for bounded analytic functions on the unit disk.
Quasiconformal Extension of Strongly Spirallike Functions Peer-reviewed

Toshiyuki Sugawa

COMPUTATIONAL METHODS AND FUNCTION THEORY　12　(1)　19-30　2012
Publisher: HELDERMANN VERLAG
ISSN：1617-9447

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We show that a strongly lambda-spirallike function of order alpha can be extended to a sin(pi alpha/2)-quasiconformal automorphism of the complex plane for -pi/2 < lambda < pi/2 and 0 < alpha < 1 with vertical bar lambda vertical bar < pi alpha/2. Towards the proof we provide several geometric characterizations of a strongly lambda-spirallike domain of order alpha. We also give a concrete form of the mapping function of the standard strongly lambda-spirallike domain U-lambda,U-alpha of order alpha. A key tool of the present study is the notion of lambda-argument, which was developed by Y. C. Kim and the author [5].
Invariant Schwarzian Derivatives of Higher Order Peer-reviewed

Seong-A Kim, Toshiyuki Sugawa

COMPLEX ANALYSIS AND OPERATOR THEORY　5　(3)　659-670　2011/09
Publisher: BIRKHAUSER VERLAG AG
DOI： 10.1007/s11785-010-0081-6 　

ISSN：1661-8254

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We derive relations between the Aharonov invariants and Tamanoi's Schwarzian derivatives of higher order and give a recursive formula for Tamanoi's Schwarzians. Then we propose a definition of invariant Schwarzian derivatives of a nonconstant holomorphic map between Riemann surfaces with conformal metrics. We show a recursive formula also for our invariant Schwarzians.
Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky Peer-reviewed

Daniela Kraus, Oliver Roth, Toshiyuki Sugawa

MATHEMATISCHE ZEITSCHRIFT　267　(3-4)　851-868　2011/04
Publisher: SPRINGER
DOI： 10.1007/s00209-009-0649-x 　

ISSN：0025-5874

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An explicit formula for the generalized hyperbolic metric on the thrice-punctured sphere with singularities of order alpha (j) a parts per thousand currency sign 1 at z (j) is obtained in all possible cases alpha (1) + alpha (2) + alpha (3) > 2. The existence and uniqueness of such a metric was proved long time ago by Picard (J Reine Angew Math 130:243-258, 1905) and Heins (Nagoya Math J 21:1-60, 1962), while explicit formulas for the cases alpha (1) = alpha (2) = 1 were given earlier by Agard (Ann Acad Sci Fenn Ser A I 413, 1968) and recently by Anderson et al. (Math Z, to appear, doi:10.1007/s00209-009-0560-5="10.1007/s00209-009-0560-5" TargetType). We also establish precise and explicit lower bounds for the generalized hyperbolic metric. This extends work of Hempel (J Lond Math Soc II Ser 20:435-445, 1979) and Minda (Complex Var 8:129-144, 1987). As applications, sharp versions of Landau- and Schottky-type theorems for meromorphic functions are obtained.
GEOMETRIC INVARIANTS ASSOCIATED WITH PROJECTIVE STRUCTURES AND UNIVALENCE CRITERIA Peer-reviewed

Seong-A Kim, Toshiyuki Sugawa

TOHOKU MATHEMATICAL JOURNAL　63　(1)　41-57　2011/03
Publisher: TOHOKU UNIVERSITY
DOI： 10.2748/tmj/1303219935 　

ISSN：0040-8735

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For a nonconstant holomorphic map between projective Riemann surfaces with conformal metrics, we consider invariant Schwarzian derivatives and projective Schwarzian derivatives of general virtual order. We show that these two quantities are related by the "Schwarzian derivative" of the metrics of the surfaces (at least for the case of virtual orders 2 and 3). As an application, we give univalence criteria for a meromorphic function on the unit disk in terms of the projective Schwarzian derivative of virtual order 3.
Sharp norm estimate of Schwarzian derivative for a class of convex functions Peer-reviewed

Stanislawa Kanas, Toshiyuki Sugawa

ANNALES POLONICI MATHEMATICI　101　(1)　75-86　2011
Publisher: POLISH ACAD SCIENCES INST MATHEMATICS
DOI： 10.4064/ap101-1-8 　

ISSN：0066-2216

eISSN：1730-6272

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We establish a sharp norm estimate of the Schwarzian derivative for a function in the classes of convex functions introduced by Ma and Minda [Proceedings of the Conference on Complex Analysis, Int. Press, 1992, 157-169]. As applications, we give sharp norm estimates for strongly convex functions of order alpha, 0 < alpha < 1, and for uniformly convex functions.
HARDY SPACES AND UNBOUNDED QUASIDISKS Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA　36　(1)　291-300　2011
Publisher: SUOMALAINEN TIEDEAKATEMIA
DOI： 10.5186/aasfm.2011.3618 　

ISSN：1239-629X

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We study the maximal number 0 <= h <= +infinity for a given plane domain Omega such that f is an element of H(p) whenever p < h and f is analytic in the unit disk with values in Omega. One of our main contributions is an estimate of h for unbounded K-quasidisks.
A conformally invariant metric on Riemann surfaces associated with integrable holomorphic quadratic differentials Peer-reviewed

Toshiyuki Sugawa

MATHEMATISCHE ZEITSCHRIFT　266　(3)　645-664　2010/11
Publisher: SPRINGER
DOI： 10.1007/s00209-009-0590-z 　

ISSN：0025-5874

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In this paper, we define a conformally invariant (pseudo-)metric on all Riemann surfaces in terms of integrable holomorphic quadratic differentials and analyze it. This metric is closely related to an extremal problem on the surface. As a result, we have a kind of reproducing formula for integrable quadratic differentials. Furthermore, we establish a new characterization of uniform thickness of hyperbolic Riemann surfaces in terms of invariant metrics.
Twice-punctured hyperbolic sphere with a conical singularity and generalized elliptic integral Peer-reviewed

G. D. Anderson, T. Sugawa, M. K. Vamanamurthy, M. Vuorinen

MATHEMATISCHE ZEITSCHRIFT　266　(1)　181-191　2010/09
Publisher: SPRINGER
DOI： 10.1007/s00209-009-0560-5 　

ISSN：0025-5874

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We describe, in terms of generalized elliptic integrals, the hyperbolic metric of the twice-punctured sphere with one conical singularity of prescribed order. We also give several monotonicity properties of the metric and a couple of applications.
Dual mean value problem for complex polynomials Peer-reviewed

Vladimir Dubinin, Toshiyuki Sugawa

PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES　85　(9)　135-137　2009/11
Publisher: JAPAN ACAD
DOI： 10.3792/pjaa.85.135 　

ISSN：0386-2194

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We consider an extremal problem for polynomials, which is dual to the well-known Smale mean value problem. We give a rough estimate depending only on the degree.
A NOTE ON BAZILEVIC FUNCTIONS Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

TAIWANESE JOURNAL OF MATHEMATICS　13　(5)　1489-1495　2009/10
Publisher: MATHEMATICAL SOC REP CHINA
ISSN：1027-5487

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We provide a few sufficient conditions for a normalized analytic function in the unit disk to be a Bazilevic function of prescribed type.
Completely monotone sequences and universally prestarlike functions Peer-reviewed

Stephan Ruscheweyh, Luis Salinas, Toshiyuki Sugawa

ISRAEL JOURNAL OF MATHEMATICS　171　(1)　285-304　2009/06
Publisher: HEBREW UNIV MAGNES PRESS
DOI： 10.1007/s11856-009-0050-9 　

ISSN：0021-2172

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We introduce universally convex, starlike and prestarlike functions in the slit domain a", [1, a), and show that there exists a very close link to completely monotone sequences and Pick functions.
Uniformly locally univalent functions and Hardy spaces Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS　353　(1)　61-67　2009/05
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI： 10.1016/j.jmaa.2008.11.036 　

ISSN：0022-247X

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We consider the class of uniformly locally univalent functions on the unit disk with prescribed pre-Schwarzian norm. In the present paper, we show that the class is contained in the Hardy space of a specific exponent depending on the norm. (C) 2008 Elsevier Inc. All rights reserved.
A coefficient inequality for Bloch functions with applications to uniformly locally univalent functions Peer-reviewed

Toshiyuki Sugawa, Takao Terada

MONATSHEFTE FUR MATHEMATIK　156　(2)　167-173　2009/02
Publisher: SPRINGER WIEN
DOI： 10.1007/s00605-008-0548-y 　

ISSN：0026-9255

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We give a Fekete-Szego type inequality for an analytic function on the unit disk with Bloch seminorm a parts per thousand currency sign1. As an application of it, we derive a sharp inequality for the third coefficient of a uniformly locally univalent function f(z) = z + a (2) z (2) + a (3) z (3) + a <- on the unit disk with pre-Schwarzian norm a parts per thousand currency sign lambda for a given lambda > 0.
NORM ESTIMATES AND UNIVALENCE CRITERIA FOR MEROMORPHIC FUNCTIONS Peer-reviewed

Saminathan Ponnusamy, Toshiyuki Sugawa

JOURNAL OF THE KOREAN MATHEMATICAL SOCIETY　45　(6)　1661-1676　2008/11
Publisher: KOREAN MATHEMATICAL SOC
ISSN：0304-9914

eISSN：2234-3008

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Norm estimates of the pre-Schwarzian derivatives are given for meromorphic functions in the outside of the unit circle. We deduce several univalence criteria for meromorphic functions from those estimates.
A note on a theorem of Chuaqui and Gevirtz Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA　33　(1)　273-279　2008
Publisher: SUOMALAINEN TIEDEAKATEMIA
ISSN：1239-629X

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For a subdomain Q of the right half-plane H, Chuaqui and Gevirtz showed the following theorem: the image f(D) of the unit disk D under ail analytic function f on D is a quasidisk whenever f'(D)subset of Omega if and only if there exists a compact subset K of H such that sK boolean AND (H\Omega)not equal 0 for any positive number s. We show that this condition is equivalent to the inequality W(Omega)< 2, where W(Omega) stands for the circular width of the domain Omega.
The limiting shape of one-dimensional Teichmuller spaces Peer-reviewed

Toshiyuki Sugawa

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY　136　(8)　2849-2858　2008
Publisher: AMER MATHEMATICAL SOC
DOI： 10.1090/S0002-9939-08-09428-8 　

ISSN：0002-9939

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We show that the Bers embedding of the Teichmuller space of a once-punctured torus converges to the cardioid in the sense of Caratheodory up to rotation when the base torus goes to the boundary of its moduli space.
Strong starlikeness for a class of convex functions Peer-reviewed

Stanislawa Kanas, Toshiyuki Sugawa

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS　336　(2)　1005-1017　2007/12
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI： 10.1016/j.jmaa.2007.03.025 　

ISSN：0022-247X

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By means of the Briot-Bouquet differential subordination, we estimate the order of strong starlikeness of strongly convex functions of a prescribed order. We also make numerical experiments to examine our estimates. (c) 2007 Elsevier Inc. All rights reserved.
A general approach to the Fekete-Szego problem Peer-reviewed

Jae Ho Choi, Yong Chan Kim, Toshiyuki Sugawa

JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN　59　(3)　707-727　2007/07
Publisher: MATH SOC JAPAN
DOI： 10.2969/jmsj/05930707 　

ISSN：0025-5645

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In this article, we provide a new method solving the Fekete-Szego problem for classes of close-to-convex functions defined in terms of subordination. As an example, we apply it to the class of strongly close-to-convex functions.
Invariant differential operators associated with a conformal metric Peer-reviewed

Seong-A Kim, Toshiyuki Sugawa

MICHIGAN MATHEMATICAL JOURNAL　55　(2)　459-479　2007
Publisher: MICHIGAN MATHEMATICAL JOURNAL
DOI： 10.1307/mmj/1187647003 　

ISSN：0026-2285
The Alexander transform of a spirallike function Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS　325　(1)　608-611　2007/01
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI： 10.1016/j.jmaa.2006.01.077 　

ISSN：0022-247X

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We show that the Alexander transform of a ss-spirallike function is univalent when cos ss <= 1/2, which settles the problem posed by Robertson. We also solved a problem considered by Y.J. Kim and Merkes. (c) 2006 Elsevier Inc. All rights reserved.
Geometric function theory and Smale's mean value conjecture Peer-reviewed

Ege Fujikawa, Toshiyuki Sugawa

PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES　82　(7)　97-100　2006/09
Publisher: JAPAN ACAD
ISSN：0386-2194

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We improve an estimate of the constant in Smale's mean value conjecture, by using the Bieberbach theorem for coefficients of univalent functions and an estimate of the hyperbolic density of a certain simply connected domain.
Norm estimates of the pre-Schwarzian derivatives for certain classes of univalent functions Peer-reviewed

YC Kim, T Sugawa

PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY　49　(Part 1)　131-143　2006/02
Publisher: CAMBRIDGE UNIV PRESS
DOI： 10.1017/S0013091504000306 　

ISSN：0013-0915

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A sharp norm estimate will be given to the pre-Schwarzian derivatives of close-to-convex functions of specified type. In order to show the sharpness, we introduce a kind of maximal operator which may be of independent interest. We also discuss a relation between the subclasses of close-to-convex functions and the Hardy spaces.
A Conformal invariant for nonvanishing analytic functions and its applications Peer-reviewed

Yong Chan Kim, Toshiyuki Sugawa

MICHIGAN MATHEMATICAL JOURNAL　54　(2)　393-410　2006
Publisher: MICHIGAN MATHEMATICAL JOURNAL
DOI： 10.1307/mmj/1156345602 　

ISSN：0026-2285
On conformal representations of the interior of an ellipse Peer-reviewed

Stanislawa Kanas, Toshiyuki Sugawa

ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA　31　(2)　329-348　2006
Publisher: SUOMALAINEN TIEDEAKATEMIA
ISSN：1239-629X

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We consider the conformal mappings f and g of the unit disk onto the inside of an ellipse with foci at 1 so that f(0) = 0, f'(0) > 0, g(0) = -1 and g'(0) > 0. The main purpose of this article is to show positivity of the Taylor coefficients of f and g about the origin. To this end, we use a special relation between f and g and the fact that f satisfies a second-order linear ODE. Some applications are given to the class of k-uniformly convex functions.
Drawing Bers embeddings of the Teichmuller space of once-punctured tori Peer-reviewed

Y Komori, T Sugawa, M Wada, Y Yamashita

EXPERIMENTAL MATHEMATICS　15　(1)　51-60　2006
Publisher: A K PETERS LTD
DOI： 10.1080/10586458.2006.10128951 　

ISSN：1058-6458

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We present a computer-oriented method of producing pictures of Bers embeddings of the Teichmuller space of once-punctured tori. The coordinate plane is chosen in such a way that the accessory parameter is hidden in the relative position of the origin. Our algorithm consists of two steps. For each point in the coordinate plane, we first compute the corresponding monodromy representation by numerical integration along certain loops. Then we decide whether the representation is discrete by applying Jorgensen's theory on the quasi-Fuchsian space of once-punctured tori.
Characterizations of hyperbolically convex regions Peer-reviewed

SA Kim, T Sugawa

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS　309　(1)　37-51　2005/09
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI： 10.1016/j.jmaa.2004.12.008 　

ISSN：0022-247X

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Let X be a simply connected and hyperbolic subregion of the complex plane C. A proper subregion Omega of X is called hyperbolically convex in X if for any two points A and B in Omega, the hyperbolic geodesic arc joining A and B in X is always contained in Omega. We establish a number of characterizations of hyperbolically convex regions Omega in X in terms of the relative hyperbolic density rho(Omega) (w) of the hyperbolic metric of Omega to X, that is the ratio of the hyperbolic metric lambda(Omega)(w) |dw| of Omega to the hyperbolic metric lambda(X)(w) |dw| of X. Introduction of hyperbolic differential operators on X makes calculations much simpler and gives analogous results to some known characterizations for euclidean or spherical convex regions. The notion of hyperbolic concavity relative to X for real-valued functions on Omega is also given to describe some sufficient conditions for hyperbolic convexity. (c) 2004 Elsevier Inc. All rights reserved.
Some inequalities for the Poincare metric of plane domains Peer-reviewed

T Sugawa, M Vuorinen

MATHEMATISCHE ZEITSCHRIFT　250　(4)　885-906　2005/08
Publisher: SPRINGER
DOI： 10.1007/s00209-005-0782-0 　

ISSN：0025-5874

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In this paper, the Poincare ( or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain C \ {0, 1}. In particular, another proof of a recent result of Gardiner and Lakic [ 7] is given with explicit constant. This and some other constants in this paper involve particular values of complete elliptic integrals and related special functions. A concrete estimate for the hyperbolic distance near a boundary point is also given, from which refinements of Littlewood's theorem are derived.
Norm estimates for the Alexander transforms of convex functions of order alpha Peer-reviewed

JH Choi, YC Kim, S Ponnusamy, T Sugawa

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS　303　(2)　661-668　2005/03
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI： 10.1016/j.jmaa.2004.08.066 　

ISSN：0022-247X

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The hyperbolic sup norm of the pre-Schwarzian derivative of a locally univalent function on the unit disk measures the deviation of the function from similarities. We present sharp norm estimates for the Alexander transforms of convex functions of order alpha, 0 less than or equal to alpha < 1. (C) 2004 Elsevier Inc. All rights reserved.
On the degenerate Beltrami equation Peer-reviewed

Gutlyanskii, V, O Martio, T Sugawa, M Vuorinen

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY　357　(3)　875-900　2005
Publisher: AMER MATHEMATICAL SOC
DOI： 10.1090/S0002-9947-04-03708-0 　

ISSN：0002-9947

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We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient mu(z) has the norm parallel tomuparallel to(infinity) = 1. Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of mu. A uniqueness theorem is also proved when the singular set Sing(mu) of mu is contained in a totally disconnected compact set with an additional thinness condition on Sing(mu).
Mapping properties of nonlinear integral operators and pre-Schwarzian derivatives Peer-reviewed

YC Kim, S Ponnusamy, T Sugawa

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS　299　(2)　433-447　2004/11
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI： 10.1016/j.jmaa.2004.03.081 　

ISSN：0022-247X

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In this paper, nonlinear integral operators on normalized analytic functions in the unit disk are investigated in connection with the pre-Schwarzian derivative and the Hornich operation. In particular, several nontrivial relations between these operators and the class of strongly starlike functions will be deduced. (C) 2004 Elsevier Inc. All rights reserved.
Geometric properties of nonlinear integral transforms of certain analytic functions Peer-reviewed

YC Kim, S Ponnusamy, T Sugawa

PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES　80　(5)　57-60　2004/05
Publisher: JAPAN ACAD
ISSN：0386-2194

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By using norm estimates of the pre-Schwarzian derivatives for certain analytic functions defined by a nonlinear integral transform, we shall give several interesting geometric properties of the integral transform.
Some counterexamples in dynamics of rational semigroups Peer-reviewed

R Stankewitz, T Sugawa, H Sumi

ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA　29　(2)　357-366　2004
Publisher: SUOMALAINEN TIEDEAKATEMIA
ISSN：1239-629X

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We give an example of two rational functions with non-equal Julia sets that generate a rational semigroup whose completely invariant Julia set is a closed line segment. We also give an example of polynomials with unequal Julia sets that generate a non nearly Abelian polynomial semigroup with the property that the Julia set of one generator is equal to the Julia set of the semigroup. These examples show that certain conjectures in the field of dynamics of rational semigroups do not hold as stated and therefore require the allowance of certain exceptional cases.
Existence and uniqueness theorems for the Beltrami equation with degeneration Peer-reviewed

M Viorinen, VY Gutlyanskii, O Martio, T Sugawa

DOKLADY MATHEMATICS　68　(3)　313-315　2003/11
Publisher: INTERPERIODICA
ISSN：1064-5624
Growth and coefficient estimates for uniformly locally univalent functions on the unit disk Peer-reviewed

Yong Chang Kim, Toshiyuki Sugawa

Rocky Mountain Journal of Mathematics　32　(1)　179-200　2002

DOI： 10.1216/rmjm/1030539616 　

ISSN：0035-7596

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In this note we shall give a sharpd istortion estimate for a uniformly locally univalent holomorphic function on the unit disk in terms of the norm of pre-Schwarzian derivative. As applications, we shall investigate the growth of coefficients and integral means of such a function and mention a connection with Hardy spaces. We also give norm estimates for typical classes of univalent functions. © 2002 Rocky Mountain Mathematics Consortium.
一様完全集合 - その解析的・幾何的側面 -

須川敏幸

数学　53　(4)　387-402　2001/10/01
Publisher: The Mathematical Society of Japan
DOI： 10.11429/sugaku1947.53.387 　

ISSN：0039-470X
An explicit bound for uniform perfectness of the Julia sets of rational maps Peer-reviewed

T Sugawa

MATHEMATISCHE ZEITSCHRIFT　238　(2)　317-333　2001/10
Publisher: SPRINGER-VERLAG
DOI： 10.1007/s002090100255 　

ISSN：0025-5874

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A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Mane and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and, as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application, we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials f(c)(z) = z(2) +c in terms of the parameter c.
Uniform perfectness of the limit sets of Kleinian groups Peer-reviewed

T Sugawa

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY　353　(9)　3603-3615　2001
Publisher: AMER MATHEMATICAL SOC
DOI： 10.1090/S0002-9947-01-02775-1 　

ISSN：0002-9947

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In this note, we show, in a quantitative fashion, that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., if the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. This result covers the known case when the group is analytically finite. As applications, we present estimates of the Hausdorff dimension of the limit set and the translation lengths in the region of discontinuity for such a Kleinian group. Several examples will also be given.
Coefficient bounds and convolution properties for certain classes of close-to-convex functions Peer-reviewed

YC Kim, JH Choi, T Sugawa

PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES　76　(6)　95-98　2000/06
Publisher: JAPAN ACAD
ISSN：0386-2194

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A number of authors (cf. Koepf [4], Ma and Minda [6]) have been studying the sharp upper bound on the coefficient functional \a(3) - mu a(2)(2) \ for certain classes of univalent functions. In this paper, we consider the class C(phi, psi) of normalized close-to-convex functions which is defined by using subordination for analytic functions phi and psi on the unit disk. Our main object is to provide bounds of the quantity a(3) - mu a(2)(2) for functions f(z) = z + a(2)z(2) + a(3)z(3) + ... in C(phi, psi) in terms of phi and psi, where mu is a real constant. We also show that the class C(phi, psi) is closed under the convolution operation by convex functions, or starlike functions of order 1/2 when phi and psi, satisfy some mild conditions.
Uniform Perfectness of the Julia Sets of Quadratic Polynomials

Science bulletin of Josai University, Special Issue　4　47-56　1998
Publisher: None
DOI： 10.20566/13429590_4_47 　

ISSN：1342-9590
Various domain constants related to uniform perfectness

Sugawa Toshiyuki

Complex Variables Theory and Application　36　(4)　311-345　1998
Publisher: Taylor & Francis
ISSN：0278-1077
On the space of schlicht projective structures on compact Riemann surfaces with boundary Peer-reviewed

T Sugawa

JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY　35　(4)　697-732　1995/12
Publisher: KINOKUNIYA CO LTD
ISSN：0023-608X
THE BERS PROJECTION AND THE LAMBDA-LEMMA Peer-reviewed

T SUGAWA

JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY　32　(4)　701-713　1992/12
Publisher: KINOKUNIYA CO LTD
ISSN：0023-608X
ON THE BERS CONJECTURE FOR FUCHSIAN-GROUPS OF THE 2ND KIND Peer-reviewed

SUGAWA T

JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY　32　(1)　45-52　1992

ISSN：0023-608X

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Misc. 21

複素解析におけるポテンシャル論—特集ポテンシャルを探る

須川敏幸

数理科学　60　(3)　35-41　2022/03
Publisher: サイエンス社
ISSN：0386-2240
書評志賀啓成 : リーマンと解析学,リーマンの生きる数学(黒川信重編),2,共立出版,2020年,ⅷ+98ページ

須川敏幸

数学 / 日本数学会編　74　(3)　331-334　2022

ISSN：0039-470X
複素双曲空間入門—特集非ユークリッド幾何学の数理

須川敏幸

数理科学　59　(4)　16-23　2021/04
Publisher: サイエンス社
ISSN：0386-2240
複素数ってそもそも何なの?—特集複素関数の質問箱

須川敏幸

数学セミナー　57　(6)　8-12　2018/06
Publisher: 日本評論社
ISSN：0386-4960
複素解析—特集分野を語る10の定理

須川敏幸

数学セミナー　56　(8)　28-32　2017/08
Publisher: 日本評論社
ISSN：0386-4960
ワイルとリーマン面—特集ワイル : 現代の数学と物理に与えた影響を探る

須川敏幸

数理科学　54　(10)　15-21　2016/10
Publisher: サイエンス社
ISSN：0386-2240
エレガントな解答をもとむ解答 : 出題 2015年2月号

須川敏幸, 徳重典英, 竹縄知之

数学セミナー　54　(5)　90-99　2015/05
Publisher: 日本評論社
ISSN：0386-4960
リーマン面の分岐被覆—特集被覆のはなし

須川敏幸

数学セミナー　52　(1)　24-29　2013/01
Publisher: 日本評論社
ISSN：0386-4960
複素関数論への招待—特集大学数学が一望できる数学ランドへようこそ(その1)

須川敏幸

数学セミナー　50　(5)　44-49　2011/05
Publisher: 日本評論社
ISSN：0386-4960
エレガントな解答をもとむ解答--出題 2010年12月号

有澤誠, 須川敏幸

数学セミナー　50　(3)　98-106　2011/03
Publisher: 日本評論社
ISSN：0386-4960
書評 E. de Faria and W. de Melo: Mathematical tools for one-dimensional dynamics

須川敏幸

数学 / 日本数学会編　63　(3)　363-366　2011
Publisher: 日本数学会
ISSN：0039-470X

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記事種別: 書評
エレガントな解答をもとむ解答--出題 2009年4月号

須川敏幸, 萩田真理子

数学セミナー　48　(7)　86-93　2009/07
Publisher: 日本評論社
ISSN：0386-4960
エレガントな解答をもとむ解答--出題 2008年4月号

須川敏幸, 中内伸光

数学セミナー　47　(7)　102-108　2008/07
Publisher: 日本評論社
ISSN：0386-4960
数列の極限—特集極限--知れば知るほど深まる数学

須川敏幸

数学セミナー　46　(6)　12-16　2007/06
Publisher: 日本評論社
ISSN：0386-4960
書評谷口雅彦:フラクタル曲線についての解析学--擬等角写像外伝

須川敏幸

数学 / 日本数学会編　59　(1)　105-108　2007/01
Publisher: 日本数学会
ISSN：0039-470X

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記事種別: 書評
ON UNIVALENCE CRITERIA FOR MEROMORPHIC FUNCTIONS (Coefficient Inequalities in Univalent Function Theory and Related Topics)

Sugawa Toshiyuki

RIMS Kokyuroku　1414　117-122　2005/02
Publisher: Kyoto University
ISSN：1880-2818
DRAWING BERS EMBEDDINGS OF THE TEICHMULLER SPACE OF ONCE PUNCTURED TORI (Hyperbolic Spaces and Related Topics II)

Komori Yohei, Sugawa Toshiyuki, Wada Masaaki, Yamashita Yasushi

RIMS Kokyuroku　1163　9-17　2000/07
Publisher: Kyoto University
ISSN：1880-2818
INNER RADIUS OF UNIVALENCE FOR A STRONGLY STARLIKE DOMAIN (New Extension of Historical Theorems for Univalent Function Theory)

Sugawa Toshiyuki

RIMS Kokyuroku　1164　144-150　2000/07
Publisher: Kyoto University
ISSN：1880-2818
On an application of ribbon graphs to resolution of the Branch singularities of Hyeperbolic 2-ORBIFOLDS (Hyperbolic Spaces and Related Topics)

Sugawa Toshiyuki

RIMS Kokyuroku　1104　25-29　1999/07
Publisher: Kyoto University
ISSN：1880-2818
A GEOMETRIC PROOF OF SELBERG'S LEMMA FOR FUCHSIAN GROUPS (Analysis and Geometry of Hyperbolic Spaces)

Sugawa Toshiyuki

RIMS Kokyuroku　1065　1-13　1998/10
Publisher: Kyoto University
ISSN：1880-2818
GROWTH AND COEFFICIENT ESTIMATES FOR UNIFORMLY LOCALLY UNIVALENT FUNCTIONS ON THE UNIT DISK (Applications of Complex Function Theory to Differential Equations)

Kim Yong Chan, Sugawa Toshiyuki

RIMS Kokyuroku　1062　34-46　1998/09
Publisher: Kyoto University
ISSN：1880-2818

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Books and Other Publications 2

複素解析学特論

楠, 幸男, 須川, 敏幸

現代数学社　2019/11

ISBN: 9784768705209
Topics in Finite or Infinite Dimensional Complex Analysis

Katsuhiko Matsuzaki, Toshiyuki Sugawa

Tohoku University Press　2013/01

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Proceedings of the 19th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, held in Hiroshima, 2011.

Research Projects 31

Quasiconformal extension in differential geometry and theory of the universal Teichmueller space in harmonic analysis

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Waseda University

2018/04/01 - 2023/03/31
Analysis of elliptic operators and its applications to Geometric Function Theory

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

Category: Grant-in-Aid for Scientific Research (B)

Institution: Tohoku University

2017/04/01 - 2022/03/31
複雑領域のポテンシャル解析の深化―非線形PDEと理想境界への応用

相川弘明, 志賀啓成, 倉田和浩, 須川敏幸, 平田賢太郎, 鈴木紀明, 正宗淳, 利根川吉廣, 木上淳, 加須栄篤, 堀田一敬, 野瀬敏洋

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 基盤研究(A)

2017/04/01 - 2021/03/31

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Intrinsic ultracontractivity を応用して，Lipschitz領域やJohn領域をベースにするシリンダー上の熱方程式の優解の可積分性を与え，正値優調和関数の可積分性を放物型に拡張した．境界条件付きの一般化された平均曲率流であるBrakke流の時間大域解存在を証明した．非線形無限ネットワークのラプラシアンを研究し，リウヴィユ性，カシミンスキー条件，Omori-Yauタイプの弱最大値原理などの同値性や正値優調和関数の最小増大度，具体的判定条件について成果を得た．パターン形成に関わるFitzHugh反応拡散系のヘテロクリニック定常解の構成とエネルギー漸近展開を行なった. 3波相互作用を持つ非線形シュレディンガー方程式系に現れる変分問題のエネルギー最小解の存在とその漸近挙動に関する知見を得た.領域の境界の一様完全性や領域の一様性の新しいポテンシャル論的特徴づけを得た．有界Lipschitz領域において，半線形楕円型方程式の正値解に対する境界Harnack原理を，Greenポテンシャルの評価と反復法を用いて示した．孤立境界特異点をもつ正値解の漸近挙動および除去可能性について考察した．自己相似集合の Ahlfors regular conformal dimension と対応する無限グラフの parabolic index の関係および自己相似集合上への p-energy の構成について研究し，parabolic index とAhlfors regular conformal dimension の間の不等式などを示した．ポテンシャル付きの二階楕円型作用素の保存則を定式化して，対応するカシミンスキー・テストを証明した．均質化法のH収束理論を多様体に拡張し，作用素の収束を実現するリーマン計量ならびに複雑な空間への収束する多様体の族の変形理論を構築した．
Degeneration and collapsing of Kleinian groups; geometry and analysis of the compactification of their defamation spaces

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

2016/04/01 - 2021/03/31
Research on the Weil-Petersson metric of infinite dimensional Teichmueller spaces

Matsuzaki Katsuhiko, YANAGISHITA Masahiro

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Waseda University

2013/04/01 - 2019/03/31

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We introduced the Teichmueller space of diffeomorphisms of the unit circle with Hoelder continuous derivatives as a subspace of the universal Teichmueller space. We provided a complex Banach manifold structure for it and proved that its topology coincides with the one induced by the Hoelder constants of the maps. We also proved that the barycentric extension induces a continuous section from the Teichmueller space of the circle diffeomorphisms with respect to this topology. Then, we considered deformations of a group of circle diffeomorphisms with Hoelder continuous derivative and showed certain rigidity under conjugation by symmetric homeomorphisms of the circle. As an application, we give a condition for such a diffeomorphism group to be conjugate to a Moebius group by a diffeomorphism of the same regularity. The strategy is to find a fixed point of the group which acts isometrically on the integrable Teichmueller space with the Weil-Petersson metric.
Toward the construction of a new kind of Teichmueller space and its analysis

SUGAWA Toshiyuki, Zhang Tanran, Wang Li-Mei

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research

Category: Grant-in-Aid for Challenging Exploratory Research

Institution: Tohoku University

2015/04/01 - 2018/03/31

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One can assign N pairs of sides to a polygon with 2N sides in the plane. Pasting each pair of sides by a complex affine map yields a Riemann surface with affine structure. Our purpose was to construct a suitable vector bundle over the Teichmueller space which describes those Riemann surfaces arising from this construction. It turns out that it is equivalent to consider the space of affine representations of the fundamental group of a fixed Riemann surface with punctures. Moreover, if the sides have the same length for each pair in the above construction, one obtains a singular Euclidean structure for the Riemann surface. Therefore, we understand that a moduli space of such objects can be understood as a section of the above vector bundle.
Potential theory on space complexity and ideal boundary

Aikawa Hiroaki, KUMAGAI TAKASHI

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Hokkaido University

2013/04/01 - 2017/03/31

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Relationships among harmonic functions, solutions to the heat equation, the Green and heat kernels and their defining domains, the influence of the space complexity to the boundary behavior were studied. They were applied to various fields such as non-smooth Euclidean domains, manifolds, varifolds, networks and fractals. In particular, new results were obtained in Harnack principle with exceptional sets, estimates of the principal frequency in terms of capacitary width, sufficient conditions for the global boundary Harnack principle based on the capacitary width of sublevel sets of the Green function, conditions for the parabolic boundary Harnack principle (Intrinsic Ultracontractivity), the critical exponent of a graph domain enjoying the global boundary Harnack principle, and the 0-1 law of the capacity density at infinity and so on.
Analysis of degenerations and singularities of Kleinian groups and complex dynamics

SHIGA Hiroshige, AIKAWA Hiroaki, SHISHIKURA Mitsuhiro, SUGAWA Toshiyuki, ASHIKAGA Tadashi, OHSHIKA Ken'ichi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Tokyo Institute of Technology

2010/04/01 - 2015/03/31

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By developing the research of the complex structure of the deformation space of a Kleinian group, we find many complex analytic properties of the deformation space. We also find a relationship between the infectivity radius of a point in the moduli space and the hyperbolic structure as the point tends to the boundary. In the complex dynamics, we find new phenomenon on the structure of invariant sets near indifferent fixed points. A progress on the boundary Harnack principle on singular domains is done and some estimates about hyperbolic metrics with cone singularities are obtained.
Geometric structure and combinatorial structure of 3-dimensional manifolds

SAKUMA MAKOTO, SHIMADA Ichiro, DOI Hideo, YASUI Koichi, HIRANOUCHI Toshiro, KAMADA Seiichi, KONO Masaharu, NIKKUNI Ryo, AKIYOSHI Hirotaka, HIRASAWA Mikami, OHSHIKA Ken'ICHI, WADA Masaaki, MIYACHI Hideki, KIN Eiko, KOBAYASHI Tsuyoshi, YAMASHITA Yasushi, MORIMOTO Kanji, NAKANISHI Toshihiro, KOMORI Yohei, SUGAWA Toshiyuki, SHACKLETON Kenneth

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Hiroshima University

2010/04/01 - 2015/03/31

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(1) Joint work with Donghi Lee: We established a variation of McShane’s identity for 2-bridge links. Moreover, we introduced the Heckoid orbifolds and proved that they are hyperbolic, and gave a systematic construction of epimorphisms from 2-bridge link groups onto Heckoid groups. Furthermore, we proved that these are the only upper-meridian pair preserving epimorphisms onto even Heckoid groups. (2) Joint work with Ken’ichi Ohshika: We proved that for a Heegarrd surface S of a 3-manifold M with high Hempel distance, a certain natural mapping class group associated with S has a natural free decomposition. We also proved that if S is of bounded combinatorics then there is a nonempty open set U of the projective measured lamination space of S, such that any simple loop in U is not null-homntopic in M and that any two distinct simple loops in U are not homotopic in M.
Various aspects on the study of Geometric Function Theory

SUGAWA Toshiyuki, SHIGA Hiroshige, YANAGIHARA Hiroshi, SAKAN Ken-ichi, MATSUZAKI Katsuhiko, FUJIKAWA Ege, MIZUTA Yoshihiro, TANIGUCHI Masahiko, FUJIWARA Koji, ABE Makoto

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

Category: Grant-in-Aid for Scientific Research (B)

Institution: Tohoku University

2010/04/01 - 2015/03/31

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Geometric Function Theory deals with problems finding relations between geometrically described (simply-connected) domains in the plane and analytic properties of conformal mappings onto them. Bieberbach conjectured that, for a conformal mapping f(z)=a_0+a_1z+a_2z^2+... normalized by a_0=0, a_1=1, the modulus |a_n| of a_n is not greater than n.This conjecture was finally proved by de Branges in his 1985 paper. In the present research project, for example, we showed that the set of conformal mappings as above with half-integral coefficients consists of exactly 21 functions. The similar set for the integral coefficients was previously known to consist of 9 functions.
Ergodic-theoretical study of dynamical systems and stochastic processes related to the theory of Teichmuller spaces

MORITA Takehiko, SUGITA Hiroshi, OHSHIKA Ken'ichi, SUMI Hiroki, MIYACHI Hideki, ISOZAKI Yasuki, NAKADA Hitoshi, YURI Michiko, TSUJII Masato, SUGAWA Toshiyuki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Osaka University

2010/04/01 - 2014/03/31

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For the candidate of the "Teichmuller Brownian motion" obtained in our preceding research, we show that its transition probability densities satisfy some useful inequalities. Unfortunately, we are not so successful that we can establish the explicit relation of its intrinsic metric with the Teichmuller metric. On the other hand, for the renormalized Ruazy-Veech-Zorich induction, we obtain a result that illustrates the validity of Chebotarev type density theorem for the distribution of hyperbolic elements of the mapping class group.
Teichmuller spaces of symmetric structures and the rigidity and fixed-point problems of quasiconformal mapping class groups

MATSUZAKI Katsuhiko, TANIGUCHI Masahiko, SUGAWA Toshiyuki, SAKAN Kenichi, SHIGA Hiroshige, NAKANISHI Toshihiro, MIYACHI Hideki, ITO Kentaro, FUJIKAWA Ege

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

2008 - 2012

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We define various classes for self-homeomorphisms of the unit circle from a viewpoint of their quasiconformal extension, and study the parameter spaces of those families, which are regarded as Teichmuller spaces. In particular, for the Teichmuller space of symmetric homeomorphisms, we consider conditions for a group acting on this space to have a fixed point in it. As an application, we give a condition for a group of diffeomorphisms to be conjugate to the canonical group action of the circle (Mobius transformations) as well as a condition for the deformation of such a group to be trivial (that is, to have rigidity).
Research on potential problems from various aspects

AIKAWA Hiroaki, SHIGA Hiroshige, KIGAMI Jun, TONEGAWA Yoshihiro, HIRATA Kentaro, UENO Kohei, SUZUKI Noriaki, KUMAGAI Takashi, SUGAWA Toshiyuki, SHIMOMURA Tetsu, MURATA Minoru, TADOKORO Yuki, KATAGATA Koh, OHONO Takao

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (A)

Institution: Hokkaido University

2008 - 2012

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Problems on important functions such as (super, sub)harmonic functions appearing in analysis, geometry and probability are referred to as potential problems. We have investigated potential problems from various view points and unveiled deep properties of important functions in connections with nonsmooth domains, fractals, manifolds and functions spaces.
Indiscrete representations of discrete groups

MAKOTO Sakuma, SEIICHI Kamada, MASAKAZU Teragaito, KEN. ICHI Ohshika, TOSHIYUKI Sugawa, YASUSHI Yamashita, HIROTAKA Akiyoshi, HIROKI Sumi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Challenging Exploratory Research

Institution: Hiroshima University

2009 - 2011

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We completely determined those simple loops on the 2-bridge spheres of 2-bridge links to be null-homotopic or peripheral in the link complements. We also completely determined when two simple loops on the 2-bridge spheres of 2-bridge links to be homotopic in the link complements. As an application of these results, we established a variation of McShane' s identity for 2-bridge links, which gives a formula to express the modulus of the cusp of a 2-bridge link in terms of the complex translation lengths of closed geodesics
Global construction of pleating coordinates of Teichmuller spaces and its applications

KOMORI Yohei, IMAYOSHI Yoichi, SAKAN Kenichi, SAKUMA Makoto, SUGAWA Toshiyuki, MIYACHI Hideki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Osaka City University

2007 - 2010

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We estimate the Teichmuller distance between Riemann surfaces X and Y which is obtained from X byθ-grafting along a simple closed geodesic C. As anapplication we estimate the Teichmuller distance between the conformal boundary andthe convex core boundary of some regular b-groups of type (1,1) one of which gives acounterexample of K=2 conjecture. Also by means of geodesic length functions, werealize Teichmuller spaces of dimension 2 and 3 as convex polyhedron in real projective spaces.
Ergodic-theoretical study of dynamical systems and stochastic processes on moduli space of curves

MORITA Takehiko, SUGITA Hiroshi, ISOZAKI Yasuki, YOSHINO Masafumi, MATSUMOTO Makoto, IWATA Koichiro, KAWASHITA Mishio, TAKIMOTO Kazuhiro, SUGAWA Toshiyuki, NAKADA Hitoshi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Hiroshima University

2007 - 2009

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In 2007, we extended a local central limit theorem for renormalized Rauzy-Veech-Zorichinductions to a wider class of functions than the class in our previous result.from 2008, we started constructing canonical diffusion processes on the teichmuller spaces with Teichmuller metric. In spite of an unexpected difficulty, we fortunately meet a candidate of the "Teichmuller Brownian motion" via Dirichlet space theory.
Loewner方程式の幾何学的函数論および確率過程への応用

須川敏幸, 柳原宏, 盛田健彦, 吉野正史, 永井敏隆, 岩田耕一郎

Offer Organization: 日本学術振興会

System: 科学研究費助成事業挑戦的萌芽研究

Category: 挑戦的萌芽研究

2007 - 2009

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本年度は,主に東北大学において幾何学的函数論やLoewner方程式に関するセミナーを学生と一緒に行い,研究を行った.特に,Lawlerによる専門書“Conformally Invariant Processes in the Plane"をテキストに選んでSLEの考察も深めた.一方,分担者の柳原は山口大学,広島大学において幾何学的函数論やポテンシャル論など関連するセミナーに出席するなどして,研究を進めた. 当初はGutlyanskii氏を招聘してLoewner方程式に関する共同研究を再開する予定であったが,氏の多忙により調整が付かず,招聘はかなわなかった.そのかわり,平成22年2月に宮城県大崎市にある東北大学鳴子会館において,SLEやLoewner方程式に関する比較的小規模の勉強会を行った.国内から約30名の参加者が集い,情報交換や知識の共有をはかることができたので,有意義な集会となった. 以上,国内での研究に加え,引き続き韓国においてSeong-A Kim氏らと高次シュワルツ微分やシュワルツ・ピックの補題の精密化など,幾何学的函数論や等角不変量に関する共同研究を行った.また,Yong Chan Kim氏とは,ハーディー空間や単葉函数に関する今日を研究を行った. さらに,6月のCMFT2010集会(アンカラ,トルコ),8月の複素解析国際会議(トゥルク,フィンランド),11月のCMRIF集会(ナウェンチュフ,ポーランド)に招かれ,研究成果について講演を行った.
Research on deformations of real and complex manifolds and variations of invariants

SHIGA Hiroshige, AIKAWA Hiroaki, SUGAWA Toshiyuki, MURATA Minoru, MATSUZAKI Katsuhiko, NOGUCHI Junjiro, MIYAJIMA Kimio, IMAYOSHI Yoichi, SHISHIKURA Mitsuhiro, SUMI Hiroki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (A)

Institution: Tokyo Institute of Technology

2005 - 2008
New development of geometric function theory focused on conformal invariants

SUGAWA Toshiyuki, MIZUTA Yoshihiro, SAKAN Ken-ichi, SHIBA Masakazu, YOSHINO Masafumi, TANIGUCHI Masahiko, SHIGA Hiroshige, MATSUZAKI Katsuhiko, NAKAHISNI Toshihiro, SHIMOMURA Tetsu

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)

Category: Grant-in-Aid for Scientific Research (B)

2005 - 2008
Researches on quasiconformal groups and the modular group of the universal Teichmuller space

MATSUZAKI Katsuhiko, TANIGUCHI Masahiko, NAKANISHI Toshihiro, SHIGA Hiroshige, SUGAWA Toshiyuki, SAKAN Kenichi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

2004 - 2007

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The Teichmuller Tare is a deformation space of the conformal structures of a Riemann surface. The quasiconformal mapping class group is a certain quotient group of the quasiconformal homeomorphisms of the Riemann surface and it acts on the Teichmuller space as the group of biholomorphic automorphisms (modular transformations). When Teichmuller spaces are finite dimensional, they are widely studied with great importance in various fields of mathirnatics. We aim to extend them to infinite dimensional Teichmuller sperms. In this research, we investigated the dynamics of the quasiconformal mapping class on the Teichmuller space. For this purpose, we also considered a certain quotient space of the Teichmuller space, which is called the asymptotic Teichmuller space. We first investigated the recurrent set for the mapping class group and proved that the periodic points are not dense in this set. This result was a foundation of our further studies on the action of elliptic modular transformation (conformal mapping classes) and the classification of the modular transformations. Our classification was based on the behavior of the orbit and we specified two classes, which have a similar nature of the modular transformations of finite dimensional Teichmuller spaces. One is a class of stationary mapping classes, and the other is a class of modular transformations that have a fixed point on the asymptotic Teichmuller space. We noticed that the action of a stationary mapping class group is stable, but also gave an example of a non-stationary mapping class group that acts discontinuously. As an extreme case, we dealt with a mapping class group that has a common fixed point on the asymptotic Teichmuller space and proved that such a group consists of countably many elements. As another topic, we studied holomorphic self-covering of Riemann surfaces. We gave a necessary condition for a hyperbolic Riemann surface to admit a (non-injective) holomorphic self-cover in terms of the corresponding Fuchisian group. Namely, if the Fuchsian group is of divergence type at the critical exponent of its Poincare series, then the Riemann surface has no self-covers. The proof used uniqueness of the Patterson-Sullivan measure and can be extended to higher dimensional cases. A holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmuller space. We investigated the dynamics of such a self-embedding and examined the distribution of isometric tangent vectors over Teichmuller space. We also extended our observation to quasiregular self-covers of Riemann surfaces.
タイヒミュラー空間上のブラウン運動と曲線のモジュライ空間の確率解析

盛田健彦, 岩田耕一郎, 川下美潮, 須川敏幸

Offer Organization: 日本学術振興会

System: 科学研究費助成事業

Category: 萌芽研究

Institution: 広島大学

2004 - 2006

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本研究全体を通して,種数2以上のタイヒミュラー計量(T-計量)によるブラウン運動(T-BM)とヴェイユ・ピーターソン計量(WP-計量)によるブラウン運動(WP-BM)のポテンシャル論的な比較を行うことが重要な目標となっている.種数1の場合には,タイヒミュラー空間は複素上半平面とみなされ,写像類群の作用はモジュラー群の作用であって,連分数変換の2回合成というただ一つ力学系の作用と軌道同値となっており,モジュラー曲面上のブラウン運動に関するポテンシャル論的性質と,測地流のエルゴード理論的性質には関連性があることも知られている.前年度(平成17年度)に引き続き,Rauzy inductionを繰り込んで得られる力学系に対して,そのエルゴード理論的挙動を調べた.既に得られている「あるクラスの繰り込まれたRauzy inductionは自然な不変測度に関してBernoulli的であり,滑らかな観測量に関しては相関係数が指数的に減衰するのみならず,常に漸近分散が正となり,非退化な中心極限定理が成立する」という結果の精密化とアーベル微分のモジュライ空間に制限したタイヒミュラー測地流がRauzy inductionの懸垂流として表現されるという事実を組み合わせることによって対応するT-BMに如何なるポテンシャル論的性質が期待できるかを検討した.また,写像類群上のランダムウォークのにDonskerの不変原理を適用してT-BMを構成する方法についても検討した.WP-BMとT-BMのポテンシャル論的な比較については十分な議論ができなかったがBufetov氏によって最近得られた繰り込まれたRauzy inductionの一種であるRauzy-Veech-Zorich inductionに関する評価を応用することによって17年度に得られた上述の結果の精密化には到達できた.
Ergodic-theoretical study of the distribution of Teichmuller closed geodesics and the dynamical zeta functions

MORITA Takehiko, YOSHINO Masafumi, MATSUMOTO Makoto, KAWASHITA Mishio, SUGAWA Toshiyuki, NAKADA Hitoshi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: Hiroshima University

2004 - 2006

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We note that the complex upper half-plane, the modular group, the modular surface are regarded as the Teichm011er space, the mapping class group, and the moduli space of curves of genus 1, respectively. Therefore it seems natural to consider the case of genus greater than 1. In the case of genus 1, it is well known that the closed geodesics of modular surface, the conjugacy classes of primitive hyperbolic elements, and the periodic orbits of two-fold iteration of the linear fractional transformations are in the natural one-to-one correspondence and the distribution of closed geodesics satisfies the prime number type theorem. The purpose of this project is to establish the same kind of results for the moduli spaces of hyperbolic curves, especially the prime number type theorem for a restrictive class of Teichmtiller closed geodesics which corresponding to the periodic orbits of the renormalized Rauzy inductions. In 2004, we try to obtain the meromorphic extensions of the dynamical zeta functions of the renormalized Rauzy inductions. We need to establish a systematic way to extend the dynamical zeta functions of hyperbolic dynamics with singularity in sufficiently wide half-plane. To this end we first treat some typical examples to find common structure of those dynamics. As a consequence we obtain a method to extends the zeta function of two-dimensional scattering open billiards without eclipse meromorphically to the domain containing the half-plane consisting of numbers of nonnegative real part by constructing the Lipschitz continuous invariant foliations. In 2005 we generalize the above results of the meromorphic extension obtained in 2004 to a class of Lipschitz continuous Markov systems with Cantor-like invariant sets and establish a way to calculate the special values at the origin for the corresponding dynamical zeta functions. In addition, we also prove the weak type local limit theorem for a class of renormalized Rauzy inductions In 2006, we notice that the recent result obtained by Bufetov can be applied to prove the prime number type theorem for a class of renormalized Rauzy-Veech-Zorich inductions which are interesting but more restrictive class of the renormalized Rauzy inductions. The paper containing the result will be submitted to the appropriate journal in the near future.
境界の密性がもたらす領域のポテンシャル論的・幾何的性質について

須川敏幸

Offer Organization: 日本学術振興会

System: 科学研究費助成事業若手研究(B)

Category: 若手研究(B)

Institution: 広島大学

2002 - 2004

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まず,既に京都大学数理解析研究所講究録1293"ポテンシャル論とその周辺"(2002),154-167,に発表済みの「ある密性条件を満たす集合の一般化容量の評価」についてはその後もその後一般距離空間について一般化するなどの改良を加えた上で,英文の論文として現在発表準備中である. また,Seong-A Kim教授との双曲的凸領域の特徴付けに関する共同研究が一旦完了し,その後はリーマン計量を持つリーマン面上の高次の不変微分作用素に関する研究を行い,これまでに知られていた作用素を一般化することに成功した上,驚くべき漸化式を発見するに至った.これについては現在発表準備中である. Yong Chan Kim教授,S.Ponnusamy教授とは単葉函数とシュワルツ微分,双曲幾何,擬等角写像との関連について共同研究を行い,部分的な結果を得た.既にいくつかの論文で結果を発表し,それ以外についても現在論文を投稿中または発表準備中である. Stanislawa Kanas教授とは単位円板から楕円の内部への等角写像について共同研究を行い,懸案であった係数の正値性予想を解決した(現在論文投稿中).またさらに,現在は強凸状函数の強星状性について微分従属式の理論を使って研究を行った(現在発表準備中). Matti Vuorinen教授とは平面領域の双曲計量,またより一般に錐特異性を持つ双曲計量についての共同研究を行った.領域の境界点の様子を用いて双曲密度や双曲距離を評価することに成功した論文が最近Mathematische Zeitschriftに受理されたほか,もう1編を投稿準備中である.
Dynamics of modular groups on infinite dimensional Teichmuller spaces

MATSUZAKI Katsuhiko, SUGAWA Toshiyuki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Ochanomizu University

2002 - 2003

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Teichmueller spaces are not homogeneous spaces and their mudular groups do not act transitively. For compact Riemann surfaces, modular groups act discontinuously, but this is not the case for infinite dimensional Teichmueller spaces. We study the moduli spaces of Riemann surafces of infinite type by considering the chaotic behavior of the action of modular groups. For a viewpoint of general topology, the moduli space is either metrizable or not of the first separation axiom. However, except for a singular part, it can possess a certain geometric structure. In this research, we characterize this stable region by hyperbolic geometric structure of a Riemann surface and construct a contracted moduli space by the completion of the stable region. Consequently, we can describe the closure of a point set in terms of the geomery of Riemann surfaces, which is a point of teh contracted module space. We considered the space of pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian derivatives of univalent functions on the unit disk which extends to quasiconformal mappings of the extended plane in order to investigate the relation between connected components of the pre-Schwarzian model of the universal Teichmueller space and classical families of univalent functions. We also investigated geometric properties of univalent functions with a prescribed growth of the Schwarzian derivative and found that they are starlike or convex according to the distance to the origin in the Bers embedding of the universal Teichmueller space.
Analysis of quasisymmetric functions and Teichmuller space

SAKAN Ken-ichi, NISHIO Masaharu, IMAYOSHI Yoichi, NAKANISHI Toshihiro, TUGAWA Toshiyuki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Osaka City University

2001 - 2003

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The head investigator Sakan published jointly with the foreign joint worker D.Partyka "On pseudo-metrics on the space of generalized quasisymmetric automorphisms of a Jordan curve" and "On Heinz's inequality ". In the former paper they introduced, with no use of quasiconformal mappings, some pseudo-metrics into the space of generalized quasisymmetric automorphisms of a Jordan curve, and discussed some applications to topological properties of the Teichmiuller pseudo-metric. In the latter paper they generalized the Heinz's result on one-to-one harmonic mappings F of the unit disc onto itself in the case where F is the Poisson integral of a sense-preserving homeomorphic self-mapping of the unit circle. As an application they inferred a version of Heinz's inequality for harmonic and quasiconformal self-mappings of the unit disc. In a paper to be. submitted, they show an asymptotically sharp variant of Heinz's inequality for harmonic and quasiconformal self-mappings of the unit disc. For the analysis of quasisymmetry, it is important to analyze the representations of maximal dilatations and so on in terms of harmonic measure and crossratio. Furthermore, it turned out that boundary dilatations, conjugate functions and Cauchy singular integrals are deeply related to the quasiconformality of harmonic extensions. Sakan has discussed about these analyses, with investigators Nishio and Yoshida from the viewpoint of potential and probability theory. Further, on extremal extensions which are quite related to boundary dilatations Sakan has contacted foreign joint workers Y.Shen, S.Wu and investigators Ohtake and Sugawa. Moreover, Sakan has discussed with investigators Imayoshi, Komori and Nakanishi about the relation of their researches on Teichmuller space and our research project.
リーマン面の双曲幾何と一様完全性

須川敏幸

Offer Organization: 日本学術振興会

System: 科学研究費助成事業奨励研究(A)

Category: 奨励研究(A)

Institution: 京都大学

1999 - 2000

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Bergman計量の類似で、与えられたリーマン面上の可積分正則二次微分を用いてこれまで知られていない等角不変な等角計量をリーマン面に導入した。よく知られているように、二乗可積分な正則一次微分(アーベル微分)に比べて可積分正則二次微分は一般的に豊富に存在し、しかもそれはリーマン面の複素構造の変形と密接に結びついていることから、リーマン面の幾何学的性質を深く反映していると考えられる。例えば、この計量が双曲計量の定数倍で評価できることとは、可積分正則二次微分が常に双曲的有界になることと同値であることが分かる。この計量と、Bergman計量、Hahn計量、双曲計量とを比較して、リーマン面における様々な評価を行った。その結果として、この計量がHahn計量と比較可能であることが、実はリーマン面が有界幾何を持つ(つまり、平面領域の場合は境界が一様完全)であることと同値であることが証明できた。これについては現在論文を投稿中である。そのほかにも、普偏Teichmuller空間や、単葉関数の挙動についてなど、幅広い研究を行った。
Geometry of webs, Hamiltonian systems and complex dynamics

NAKAI Isao, SATO Hajime, YOSHINO Masatomi, SUWA Tatsuo, MATSUMOTO Shigenori, KIKKAWA Michihika

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B).

1998 - 2000

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We defined a canonical affine connection for integrable first order partial differential equations of finite type, and investigated its singularities along the locus where the foliation structure of solutions degenerates. We proved there exists one to one correspondence of the moduli space of equations and the space of curvature forms in some cases. We proved also the triangulability of the mapping cylinder of generic smooth mappings of manifolds. As an application we proved a naturality formula of the stiefel-homology functor of constructible functions on manifolds to homology classes.
Various quasiconformal extensions of a quasisymmetric automorphism of the unit circle and Teichmuller space

SAKAN Kenichi, KOMORI Yohei, NISHIO Masaharu, IMAYOSHI Masaharu, NAKANISHI Toshihiro, SUGAWA Toshiyuki

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (C)

Institution: Osaka City University

1998 - 1999

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1. The head investigator Sakan obtained jointly with Partyka and Zajac the following results in three published paper. (1) They gave necessary and sufficient conditions on sense-preserving homeomorphisms of the unit circle for the quasiconformality of their harmonic extensions. In particular, the case where the harmonic extension is not quasiconformal was studied. In consequence, suitable examples were constructed. (2) Suppose that the harmonic extension of a sense-preserving homeomorphism of the unit circle is quasiconformal. All such homeomorphisms with a bounded derivative were well characterized. In consequence, a generalization of Martio's result was obtained. (3) By the use of the extension operators, a unified summary on harmonic and quasiconformal extensions was given. 2. Sakan and an investigator Sugawa attended the 12th Conference on Analytic Functions (August30〜September 5, 1998 at Lublin, Poland) and gave lectures. The papers of the results were published. Sakan expressed the Beurling-Ahlfors condition on quasisymmetry in terms of harmonic measure and cross ration. In consequence, Sakan introduced a generalized conformally invariant dilatation on quasisymmetry and discussed its properties. By an application of holomorphic families of univalent functions, Sugawa obtained some results on quasiconformal extendability of univalent functions. 3. Sakan attended the Second ISAAC Conference (August 16〜21, 1999 at Fukuoka Institute of Technology) and Korea-Japan Seminar on Complex Analysis (October 18〜20, 1999 at Yeungnam University, Korea) and gave lectures. The result will soon appear in the Proceedings. On the space of all quasisymmetric automorphisms of a given Jordan curve, a conformally invariant pseudo-metric which is equivalent to the Teichmuller pseudo-metric was introduced with no use of quasiconformal extensions.
Teichmuller空間の境界の幾何学

須川敏幸

Offer Organization: 日本学術振興会

System: 科学研究費助成事業奨励研究(A)

Category: 奨励研究(A)

Institution: 京都大学

1997 - 1998

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1. 普遍タイヒミュラー空間の前Schwarz微分モデルの構造の解析。 Yong Chan Kim氏との共同研究により、前Schwarz微分のノルムからタイヒミュラー空間の点を表現する単葉函数の増大度評価や計数評価などがどのように得られるかを明らかにした(プレプリント)。2. (1,1)型タイヒミュラー空間のBers埋め込みのグラフィック化。大阪市大の小森洋平氏との共同研究により、(1,1)型リーマン面のタイヒミュラー空間のBersモデルの境界をコンピュータにより描画するための実行可能な計算原理を見いだした。実際にはある種の有理函数を係数に持つ微分方程式のモノドロミーを計算し、それを用いてBers埋め込みの境界を知るという方法であり、これまでにこのような計算例は報告されていない。今後はこれをプログラムに乗せて計算を実行させる積もりである。 3. Laplacianに関するDirichlet問題の解の境界正則性の研究。かなり緩やかな仮定の下で平面領域がLaplacianに関するDirichlet問題についてある指数以下のヘルダー連続性を保存するという結果を示した。例えば、境界が一様完全であればこの結果は正しく、全不連結な境界についてはこれまでこの種の結果は知られていなかったと思われる。同様に、Green函数の境界挙動についても詳しい評価が得られた(プレプリント)。
Analysis and Applications of Teichmuller space

TANIGUCHI Masahiko, SUGAWA Toshiyuki, KOKUBU Hiroshi, OKAJI Takashi, NOMURA Takaaki, HIRAI Takeshi

Offer Organization: Japan Society for the Promotion of Science

System: Grants-in-Aid for Scientific Research

Category: Grant-in-Aid for Scientific Research (B)

Institution: KYOTO UNIVERSITY

1996 - 1996

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The results can be devided into two categories ; those on the Teichmuller spaces and those on objects which provide the Teichmuller spaces. Among oter things, the research on the Teichmuller spaces of infinite dimension is very important. The Head investigator Taniguchi has clarified the fundamental structures of the Teichmuller space of a transcendental entire function, including the relationship between the absence of wandering domains and finite dimensionality of the corresponding Teichmuller space. These results are the coproduct with T.Harada, a graduate student, and very important, for they give a new light for the investigations on the complex dynamics induced by transcendental entire functions as in the cases of rational functions and of Kleinian groups. Also Taniguchi has investigated the coiling property, appeared only in the case of infinite dimension, and proved that the Bloch convergence on the universal Teichmuller space is equivalent to Caratheodory convergence, when we consider points of the universal teichmuller space as fractal sets on the plane. Next, a crucial divice for the Teichmuller theory is a quasiconformal map. Investigator Sugawa has suceeded to give a characterization of the Teichmuller spaces of finite Riemann surfaces without cusps. Sugawa has also given quantative estimate on domain constants related to uniform perfectness. As a consequence, we have interesting estimates of the Hausdorff dimensions of the Julia sets of a complex dynamics, or of the limit set of a Kleinian group.
λ-lemmaによるリーマン面の正則族の研究

須川敏幸

Offer Organization: 日本学術振興会

System: 科学研究費助成事業奨励研究(A)

Category: 奨励研究(A)

Institution: 京都大学

1994 - 1994

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この研究においては主にλ-lemmaの応用について考察を行った。その結果、以前から懸案であったTeichmuller空間のBers埋め込みに関する次の予想:IntS(Γ)=T(Γ)を特にΓが有限生成純双曲的第2種Fuchs群の場合について証明することが出来た。証明のポイントはIntS(Γ)の元に対応する領域を擬等角写像を用いて変形することにあるが、その時にT(Γ)に入らないとすると矛盾が出ることを示すのにIntS(Γ)の(局所)擬等角等質性が用いられ、その部分でλ-lemmaが本質的に用いられる。この証明はGehringによって既に証明されていた最も単純な場合:Γ=1の結果の別証明にもなっている。直観的には擬等角写像を用いた変形の方が分かりやすいように筆者には思われる。(この結果は“On the space of schlicht projective structures on compact Riemann surfaces with boundary"というタイトルで現在投稿中である。) なお、当初の目標であったより一般のリーマン面の正則族についてはまだ研究の緒についたばかりでまだそれほど重要な結果は得られていないが、上記の研究の副産物として双曲的平面領域の解析的普遍被覆写像のSchwarz微分の双曲的ノルムが有限であるという性質と“uniformly perfect"という性質との間に著しい関係があることを見いだした。これらの性質の同値性については既にPommerenkeが指摘しているが、筆者はより精密に定量的な評価を与えることに成功した。

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