Toshiyuki Sugawa

Toshiyuki Sugawa
Tohoku University | Tohokudai

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122
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1,396
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Introduction
Skills and Expertise
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April 2008 - present
Tohoku University
Position
  • Professor (Full)

Publications

Publications (122)
Article
Full-text available
In our previous paper (Golberg et al. in Comput Methods Funct Theory 20(3–4):539–558, 2020), we proved that the complementary components of a ring domain in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\odd...
Preprint
In our previous paper [GSV2020], we proved that the complementary components of a ring domain in $\mathbb{R}^n$ with large enough modulus may be separated by an annular ring domain and applied this result to boundary correspondence problems under quasiconformal mappings. In the present paper, we continue this work and investigate boundary extension...
Preprint
P\'olya in 1926 showed that the hypergeometric function $F(z)=\null_2F_1(a,b;c;z)$ has a totally monotone sequence as its coefficients; that is, $F$ is the generating function of a Hausdorff moment sequence, when $0\le a\le 1$ and $0\le b\le c.$ In this paper, we give a complete characterization of such hypergeometric functions $F$ in terms of comp...
Article
Full-text available
Using the definition of uniformly perfect sets in terms of convergent sequences, we apply lower bounds for the Hausdorff content of a uniformly perfect subset E of $$\mathbb {R}^n$$ R n to prove new explicit lower bounds for the Hausdorff dimension of E . These results also yield lower bounds for capacity test functions, which we introduce, and ena...
Chapter
A Kleinian group divides the Riemann sphere into two parts, the region of discontinuity and the limit set. We are interested in analytic properties of these sets from the view-point of geometric function theory.
Preprint
In the present paper, we study the shifted hypergeometric function $f(z)=z\Gauss(a,b;c;z)$ for real parameters with $0<a\le b\le c$ and its variant $g(z)=z\Gauss(a,b;c;z^2).$ Our first purpose is to solve the range problems for $f$ and $g$ posed by Ponnusamy and Vuorinen in their 2001 paper. Ruscheweyh, Salinas and Sugawa developed in their 2009 pa...
Article
Full-text available
R. Küstner proved in his 2002 paper that the function wa,b,c(z)=2F1(a+1,b;c;z)2F1(a,b;c;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w_{a,b,c}(z)=...
Preprint
Full-text available
Using the definition of uniformly perfect sets in terms of convergent sequences, we apply lower bounds for the Hausdorff content of a uniformly perfect subset $E$ of $\mathbb{R}^n$ to prove new explicit lower bounds for the Hausdorff dimension of $E.$ These results also yield lower bounds for capacity test functions, which we introduce, and enable...
Article
For a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M -condition [ 35 ]. Next, we prove that $\partial G$ is un...
Article
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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 funct...
Article
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Given a nonempty compact set \( E \) in a proper subdomain \( \Omega \) of the complex plane, we denote the diameter of \( E \) and the distance from \( E \) to the boundary of \( \Omega \) by \( d(E) \) and \( d(E,\partial\Omega) \), respectively. The quantity \( d(E)/d(E,\partial\Omega) \) is invariant under similarities and plays an important ro...
Article
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 Bi...
Article
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...
Preprint
Important geometric or analytic properties of domains in the Euclidean space $\mathbb{R}^n$ or its one-point compactification (the M\"obius space) $\overline{\mathbb{R}}^n$ $(n\ge 2)$ are often characterized by comparison inequalities between two intrinsic metrics on a domain. For instance, a proper subdomain $G$ of $\mathbb{R}^n$ is {\it uniform}...
Preprint
Full-text available
For a non-empty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega),$ respectively. The quantity $d(E)/d(E,\partial\Omega)$ is invariant under similarities and plays an important role in Geometric Function Theory....
Article
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 t...
Article
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We will provide sufficient conditions for the shifted hypergeometric function z2F1(a,b;c;z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_2F_1(a,b;c;z)$$\end{documen...
Article
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The length of the shortest closed geodesic in a hyperbolic surface $X$ is called the systole of $X.$ When $X$ is an $n$-times punctured sphere $\hat{ \mathbb{C}} \setminus A$ where $A \subset \hat{\mathbb{C}}$ is a finite set of cardinality $n\ge4,$ we define a quantity $Q(A)$ in terms of cross ratios of quadruples in $A$ so that $Q(A)$ is quantita...
Preprint
The length of the shortest closed geodesic in a hyperbolic surface $X$ is called the systole of $X.$ When $X$ is an $n$-times punctured sphere $\hat{ \mathbb{C}} \setminus A$ where $A \subset \hat{\mathbb{C}}$ is a finite set of cardinality $n\ge4,$ we define a quantity $Q(A)$ in terms of cross ratios of quadruples in $A$ so that $Q(A)$ is quantita...
Article
Full-text available
In this note, we investigate the supremum and the infimum of the functional |an+1|-|an|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|a_{n+1}|-|a_{n}|$$\end{document}...
Article
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We define a distance function on the bordered punctured disk $0<|z|\le 1/e$ 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 compara...
Article
A measurable function $\mu$ on the unit disk $\mathbb{D}$ of the complex plane with $\|\mu\|_\infty<1$ is sometimes called a Beltrami coefficient. We say that $\mu$ is trivial if it is the complex dilatation $f_{\bar z}/f_z$ of a quasiconformal automorphism $f$ of $\mathbb{D}$ satisfying the trivial boundary condition $f(z)=z,~|z|=1.$ Since it is n...
Article
Let \(\Omega \) be a domain in \({\mathbb C}\) with hyperbolic metric \(\lambda _\Omega (z)|dz|\) with Gaussian curvature \(-4.\) Mejía and Minda proved in their 1990 paper that \(\Omega \) is (Euclidean) convex if and only if \(d(z,\partial \Omega )\lambda _\Omega (z)\ge 1/2\) for \(z\in \Omega ,\) where \(d(z,\partial \Omega )\) denotes the Eucli...
Article
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 an example of a compact set in the p...
Preprint
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 an example of a compact set in the p...
Article
Full-text available
In the present paper, we study spirallikenss (including starlikeness) of the shifted hypergeometric function $f(z)=z_2F_1(a,b;c;z)$ with complex parameters $a,b,c,$ where $_2F_1(a,b;c;z)$ stands for the Gaussian hypergeometric function. First, we observe the asymptotic behaviour of $_2F_1(a,b;c;z)$ around the point $z=1$ to obtain necessary conditi...
Article
In this note, 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\H{o} problem for normalized concave functions with a prescribed pole in the unit disk.
Article
In the present paper, we will discuss the Hankel determinants $H(f) =a_2a_4-a_3^2$ of order 2 for normalized concave functions $f(z)=z+a_2z^2+a_3z^3+\dots$ with a pole at $p\in(0,1).$ Here, a meromorphic function is called concave if it maps the unit disk conformally onto a domain whose complement is convex. To this end, we will characterize the co...
Article
Full-text available
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 given analytic function $h(z)$ on the unit disk $|z|<1$ in the complex plane with $h(0)=1.$ In particular, we investigate the possible largest constant $c>0$ such that the co...
Preprint
In this note, 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\H{o} problem for normalized concave functions with a prescribed pole in the unit disk.
Article
It is well known that a hyperbolic domain in the complex plane has uniformly perfect boundary precisely when the product of its hyperbolic density and the distance function to its boundary has a positive lower bound. We extend this characterization to a hyperbolic domain in the Riemann sphere in terms of the spherical metric.
Article
Full-text available
Miller and Mocanu proved in their 1997 paper a greatly useful result which is now known as the Open Door Lemma. It provides a sufficient condition for an analytic function on the unit disk to have positive real part. Kuroki and Owa modified the lemma when the initial point is non-real. In the present note, by extending their methods, we give a suff...
Article
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.
Article
In this note, we consider meromorphic univalent functions $f(z)$ in the unit disc with a simple pole at $z=p\in(0,1)$ which have a $k$-quasiconformal extension to the extended complex plane $\hat{\mathbb C},$ where $0\leq k < 1$. We denote the class of such functions by $\Sigma_k(p)$. We first prove an area theorem for functions in this class. Next...
Article
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Marx and Strohh\"acker showed around in 1933 that $f(z)/z$ is subordinate to $1/(1-z)$ for a normalized convex function $f$ on the unit disk $|z|<1.$ Brickman, Hallenbeck, MacGregor and Wilken proved in 1973 further that $f(z)/z$ is subordinate to $k_\alpha(z)/z$ if $f$ is convex of order $\alpha$ for $1/2\le\alpha<1$ and conjectured that this is t...
Article
Let \$p(z)= z{f}^{\prime } (z)/ f(z)\$ for a function \$f(z)\$ analytic on the unit disc \$\mid z\mid \lt 1\$ in the complex plane and normalised by \$f(0)= 0, {f}^{\prime } (0)= 1\$. We provide lower and upper bounds for the best constants \${\delta }_{0} \$ and \${\delta }_{1} \$ such that the conditions \${e}^{- {\delta }_{0} / 2} \lt \mid p(z)\...
Article
The authors mainly concern the set U f of c ∈ ℂ such that the power deformation $ z(\frac{{f(z)}} {z})^c $ is univalent in the unit disk |z| < 1 for a given analytic univalent function f(z) = z + a 2z 2 + … in the unit disk. It is shown that U f is a compact, polynomially convex subset of the complex plane ℂ unless f is the identity function. In...
Article
Let $p(z)=zf'(z)/f(z)$ for a function $f(z)$ analytic on the unit disk $|z|<1$ in the complex plane and normalized by $f(0)=0, f'(0)=1.$ We will provide lower and upper bounds for the best constants $\delta_0$ and $\delta_1$ such that the conditions $e^{-\delta_0/2}<|p(z)|<e^{\delta_0/2}$ and $|p(w)/p(z)|<e^{\delta_1}$ for $|z|,|w|<1$ respectively...
Article
B. Friedman found 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 twelve functions in addition to the nine. We...
Article
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Some of important univalence criteria for a non-constant meromorphic function $f(z)$ on the unit disk $\ID$ involve its pre-Schwarzian or Schwarzian derivative. We consider an appropriate norm for the pre-Schwarzian derivative, and discuss the problem of finding the largest possible $r\in (0,1)$ for which the pre-Schwarzian norm of the dilation $r^...
Article
Let $\lambda$ be a real number with $-\pi/2<\lambda<\pi/2.$ In order to study $\lambda$-spirallike functions, it is natural to measure the angle according to $\lambda$-spirals. Thus we are led to the notion of $\lambda$-argument. This fits well the classical correspondence between $\lambda$-spirallike functions and starlike functions. Using this id...
Article
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.
Article
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M. Biernacki gave 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 $|z|<1$ in 1936. 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 $|z|...
Article
We study the boundary correspondence under $\mu$-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 $f$...
Article
Full-text available
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 Schur's continued fraction algorithm and several inequalities for bounded analytic funct...
Article
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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, International Press Inc., 1992, 157-169]. As applications, we give sharp norm estimates for strongly convex functions of order $\alpha,~0<\alpha<1,$ and for...
Article
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 $|\lambda|<\pi\alpha/2.$ In order to prove it, we provide several geometric characterizations of a strongly $\lambda$-spirallike domain of...
Article
For an analytic function $f(z)$ on the unit disk $|z|<1$ with $f(0)=f'(0)-1=0$ and $f(z)\ne0, 0<|z|<1,$ 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 $f\mapsto f_c$ maps a specified class of univalent functions into the class of univalent functions. A little sur...
Article
We study the maximal number 0≤h≤+∞ for a given plane domain Ω such that f∈H p whenever p<h and f is analytic in the unit disk with values in Ω. One of our main contributions is an estimate of h for unbounded K-quasidisks.
Article
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...
Article
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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.
Article
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This is an expository account on quasiconformal mappings and µ-conformal homeomorphisms with an emphasis on the role played by the modulus of an annulus or a semiannulus. In order that the reader gets acquainted with modulus techniques, we give proofs for some of typical and important results. We also include several recent results on µ-conformal h...
Article
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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 order...
Article
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We argue relations between the Aharonov invariants and Tamanoi's Schwarzian derivatives of higher order and give a recursion 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 recursion fomula also for our inv...
Article
An explicit formula for the generalized hyperbolic metric on the thrice-punctured sphere \({\mathbb {P} \backslash \{z_1, z_2, z_3\}}\) with singularities of order α j ≤ 1 at z j is obtained in all possible cases α 1 + α 2 + α 3 > 2. The existence and uniqueness of such a metric was proved long time ago by Picard (J Reine Angew Math 130:243–258, 19...
Article
We provide a few sufficient conditions for a normalized analytic function in the unit disk to be a Bazilevič function.
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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. Comment: 4 pages
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We introduce universally convex, starlike and prestarlike functions in the slit domain ℂ [1, ∞), and show that there exists a very close link to completely monotone sequences and Pick functions.
Article
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.
Article
We give a Fekete-Szegö type inequality for an analytic function on the unit disk with Bloch seminorm ≤1. 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 + ⋯ on the unit disk with pre-Schwarzian norm ≤λ for a given λ > 0.
Article
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Norm estimates of the pre-Schwarzian derivatives are given for meromorphic functions in the outside of the unit circle. We de-duce several univalence criteria for meromorphic functions from those estimates.
Article
We show that the Bers embedding of the Teichmüller space of a once-punctured torus converges to the cardioid in the sense of Carathéodory up to rotation when the base torus goes to the boundary of its moduli space.
Article
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We obtain new inequalities for certain hypergeometric functions. Using these inequalities, we deduce estimates for the hyperbolic metric and the induced distance function on a certain canonical hyperbolic plane domain.
Article
For a subdomain Ω of the right half-plane H; Chuaqui and Gevirtz showed the following theorem: the image f(D) of the unit disk D under an analytic function f on D is a quasidisk whenever f″(D) ⊂ Ω if and only if there exists a compact subset K of H such that sK ∩ (H \ Ω) ≠ φ for any positive number s: We show that this condition is equivalent to th...
Article
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.
Article
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...
Article
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Peschl defined invariant higher-order derivatives of a holomorphic or mero- morphic function on the unit disk. Here, the invariance is concerned with the hyperbolic metric of the source domain and the canonical metric of the target domain. Minda and Schippers extended Peschl's invariant derivatives to the case of general conformal metrics. We intro...
Article
In this article, we provide a new method solving the Fekete-Szegö 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.
Article
We show that the Alexander transform of a β-spirallike function is univalent when cosβ⩽1/2, which settles the problem posed by Robertson. We also solved a problem considered by Y.J. Kim and Merkes.
Article
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.
Article
We give norm estimates of the pre-Schwarzian derivatives for close-to-convex functions of some special type. To this end, we i n vestigate the quantity 1 ,jzj 2 j' 0 z='zj for a zero-free holomorphic function. We also discuss a relation between the subclasses of close-to-convex functions and the Hardy spaces.
Article
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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 satis...
Article
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We present a computer-oriented method of producing pictures of Bers embeddings of the Teichmüller 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...
Article
Let X be a simply connected and hyperbolic subregion of the complex plane C. A proper subregion Ω of X is called hyperbolically convex in X if for any two points A and B in Ω, the hyperbolic geodesic arc joining A and B in X is always contained in Ω. We establish a number of characterizations of hyperbolically convex regions Ω in X in terms of the...
Article
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In this paper, the Poincaré (or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain 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 particul...
Article
We discuss the dynamics of rational semigroups, an extension of the Fatou-Julia theory of iteration of a rational map defined on the Riemann sphere. Specifically, we give some counterexamples to some conjectures relating to completely invariant Julia sets and nearly Abelian polynomial semigroups. We then state the modified conjectures as open probl...
Article
In this note, we will show that a simply connected bounded domain D is strongly starlike of order α < 1 with respect to the origin if and only if so is D<sup>∨</sup>, where D<sup>∨</sup> is the analytic inversion of the exterior of D, namely, $D^\vee=\{w\in\mathbb{C}:\, 1/w\in\widehat{\mathbb{C}} \setminus\bar D\}$ . This fact neatly explains the r...
Article
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 α, 0⩽α1.
Article
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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...
Article
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.
Article
We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient μ ( z ) \mu (z) has the norm ‖ μ ‖ ∞ = 1. \|\mu \|_\infty =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...
Article
In this note, we present a method of computing monodromies of projective structures on a once-punctured torus. This leads to an algorithm numerically visualizing the shape of the Bers embedding of a one-dimensional Teichmueller space. As a by-product, the value of the accessory parameter of a four-times punctured sphere will be calculated in a nume...
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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.
Article
This short note is a summary of the forthcoming paper (9) of the author. We present a lower estimate of the Hausdor content for a closed set with some density condition in a metric space. As an application, we give some estimate of generalized capacity for those sets.
Article
We provide an approach to the proof of positivity of the Taylor coefficients for a given conformal map of the unit disk onto a plane domain. This short note is a summary of the joint work [2] with Staniss lawa Kanas.
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We consider a second-order linear homogenous ordinary differential equations in connection with the Teichmüller space of a four-times punctured sphere. Interests will be focused on the mysterious relation between the shape of the Teichmüller space and the Fibonacci sequence. The monodromy homomorphism induced by the differential equations plays a d...
Article
The analytical theory of quasi-conformal mappings on a complex plane C implies investigating homeomorphic generalized solutions of Beltrami equation (BE) with a measurable complex-valued coefficient μ. In degenerate case, if |μ(z)|<1 for nearly all z∈C and ∥μ∥∞=ess sup |μ(z)|=1, BE may have no homeomorphic solutions, and in the case of such solutio...
Article
 The inner radius of univalence of a domain D with Poincaré density ρ D is the possible largest number σ such that the condition ∥ S f ∥ D  = sup w∈ D ρ D (w) −2∥ S f (z) ∥ ≤ σ implies univalence of f for a nonconstant meromorphic function f on D, where S f is the Schwarzian derivative of f. In this note, we give a lower bound of the inner rad...
Article
In this note we shall give a sharp distortion 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 estimat...
Article
In this note, we will prove the Ahlfors{Lehto univalence criterion in a general form. This enables us to deduce lower estimates of the inner radius of univalence for an arbitrary quasidisk in terms of a given quasiconformal re∞ection.
Article
We study the local growth of quasiconformal mappings in the plane. Esti-mates are given in terms of integral means of the pointwise angular dilatations. New sufficient conditions for a quasiconformal mapping f to be either Lipschitz or weakly Lipschitz continuous at a point are given.
Article
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 $\mathcal{C}(\varphi, \psi)$ of normalized close-to-convex functions which is defined by using subordination for analytic fu...
Article
In the unit disk D = {z ∈ ℂ ∣z∣ < 1}, the Poincaré metric ∣dz∣/(1 —∣z∣2) plays quite an important role in modern function theory. Various generalizations of this metric to Riemann surfaces have been given, e.g., hyperbolic (Kobayashi) metric, Carathéodory metric, Hahn metric, Begman metric and so on. In this article, we will provide a unifying trea...
Article
Full-text available
We present a computer-oriented method of producing pictures of Bers em- beddings 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. To each point in the coordinate plane, we flrst com- pute t...

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