arXiv subjects
Publications and source records attributed to Toshiyuki Sugawa.
We introduce the chordal isometric circle of a Möbius map, and use this to give a factorization of any Möbius map as the composition of a chordal isometry and either a reflection, or a rotary reflection, across a circle. We then use this to find the chordal, and spherical, Lipschitz constants of a Möbius map, and compare this with related results in the literature.
By a result of F. R. Keogh in 1959, we have the inequality $L(r,f)/M(r,f)\le 2π+4\log[(1+r)/(1-r)]$ for a starlike function $f$ on the unit disk, where $L(r,f)$ and $M(r,f)$ denote the length of the curve $θ\mapsto f(re^{iθ})$ and the maximum value of $|f(z)|$ on the circle $|z|=r$ for $0<r<1.$ E. Crane and D. Markose in their 2005 paper showed that the constant $4$ is best possible. D. K. Thomas obtained in 1968 the inequality $L(r,f)/\sqrt{A(r,f)}\le 2\sqrtπ(1+\log[(1+r)/(1-r)])$ for a starlike function $f$, where $A(r,f)$ is the area of the image of the disk $|z|<r$ under the mapping $f.$ In this paper, we extend these results to the case of $λ$-spirallike functions for a given $λ$ with $-π/2<λ<π/2.$
We explore the phenomenon of cavitation in higher-dimensional elasticity, defining it as the mapping of a punctured ball onto a non-degenerate ring domain. Crucially, for the class of locally quasiconformal mappings (or more general mappings) defined on the punctured ball $0<|x|<1$ in $\mathbb R^n$ that we examine, cavitation is equivalent to a failure of continuous extension to the origin. While existing modulus estimates prove insufficient for reliably detecting cavitation in this setting, our study establishes refined modulus bounds. This is achieved by introducing a novel directional dilatation which, in conjunction with the known angular dilatation, overcomes the limitations of previous methods. We illustrate our theoretical findings with several examples that demonstrate both cavitation occurrence and its absence.
Let $f$ be a meromorphic univalent function on the open unit disk having a simple pole at $p\in (0,1)$ that extends continuously to the left half $\IT^{-}$ of the unit circle. In this article, we prove that the ratio of the length of the image of the vertical diameter $\IA$ of the unit disk to the length of the image of $\IT^{-}$ under the mapping $f$ is bounded by a constant depending only on $p.$ Next, we extend this result by considering any hyperbolic geodesic and any Jordan curve in $\D$ sharing the same endpoints. These results extend the classical Gehring-Hayman inequality to meromorphic univalent functions and also prove a conjecture posed by Bhowmik and Maity [Bull. Sci. Math. \textbf{199} (2025), \# 103583].
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 problems for a larger class of mappings.
Pólya 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 complex parameters $a,b,c.$ To this end, we study the class of general properties of generating functions of Hausdorff moment sequences and, in particular, we provide a sufficient condition for the class by making use of a Phragmèn-Lindelöf type theorem. As an application, we give also a necessary and sufficient condition for a shifted hypergeometric function to be universally starlike.
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 us to characterize domains of $\mathbb{R}^n$ with uniformly perfect boundaries. Moreover, we show that an alternative method to define capacity test functions can be based on the Whitney decomposition of the domain considered.
We introduce and study invariant differential operators acting on the space $\mathcal{H}(Ω)$ of holomorphic functions on the complement ${Ω=\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w \not=1\}}$ of the "complexified unit circle" $\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w =1\}$. We obtain recursion identities, describe the behaviour under change of coordinates and find the generators of the corresponding operator algebra. We illustrate how this provides a unified framework for investigating conformally invariant differential operators on the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$, which have been studied by Peschl, Aharonov, Minda and many others, within their conjecturally natural habitat. We apply the machinery to a problem in deformation quantization by deriving explicit formulas for the canonical Wick-type star products on $Ω$, the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$ in terms of such invariant differential operators. These formulas are given in form of factorial series which depend holomorphically on a complex deformation parameter $\hbar$ and lead to asymptotic expansions of the star products in powers of $\hbar$.
In this article we prove that nonlinear resolvents of infinitesimal generators on bounded and convex subdomains of $\C^n$ 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.
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 paper the theory of universal prestarlike functions on the slit domain $\C\setminus[1,+\infty)$ and showed universal starlikeness of $f$ under some assumptions on the parameters. However, there has been no systematic study of universal convexity of the shifted hypergeometric functions except for the case $b=1.$ Our second purpose is to show universal convexity of $f$ under certain conditions on the parameters.
For a domain $G$ in the one-point compactification $\overline{\mathbb{R}}^n = \mathbb{R}^n \cup \{ \infty\}$ of $\mathbb{R}^n, n \ge 2$, we characterize the completeness of the modulus metric $μ_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio's $M$-condition. Next, we prove that $\partial G$ is uniformly perfect if and only if $μ_G$ admits a minorant in terms of a Möbius invariant metric. Several applications to quasiconformal maps are given.
R. Küstner proved in his 2002 paper that the function $w_{a,b,c}(z)=$ $F(a+1,b;c;z)/F(a,b;c;z)$ maps the unit disk $|z|<1$ onto a domain convex in the direction of the imaginary axis under some condition on the real parameters $a,b,c.$ Here $F(a,b;c;z)$ stands for the Gaussian hypergeometric function. In this paper, we study the order of convexity of $w_{a,b,c}.$ In particular, we partially solve the problem raised by the afore-mentioned paper by Küstner.
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. In particular, we study the special case of an isosceles trapezoidal polygon $L$ and obtain some estimates for the coefficient of quasiconformal reflection over $L$ in terms of special functions and geometric parameters of~$L$.
Harmonic functions are natural generalizations of conformal mappings. In recent years, a lot of work have been done by some researchers who focus on harmonic starlike functions. In this paper, we aim to introduce two classes of harmonic univalent functions of the unit disk, called hereditarily $λ$-spirallike functions and hereditarily strongly starlike functions, which are the generalizations of $λ$-spirallike functions and strongly starlike functions, respectively. We note that a relation can be obtained between this two classes. We also investigate analytic characterization of hereditarily spirallike functions and uniform boundedness of hereditarily strongly starlike functions. Some coefficient conditions are given for hereditary strong starlikeness and hereditary spirallikeness. As a simple application, we consider a special form of harmonic functions.
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.
For a non-empty compact set $E$ in a proper subdomain $Ω$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $Ω$ by $d(E)$ and $d(E,\partialΩ),$ respectively. The quantity $d(E)/d(E,\partialΩ)$ is invariant under similarities and plays an important role in Geometric Function Theory. In the present paper, when $Ω$ has the hyperbolic distance $h_Ω(z,w),$ we consider the infimum $κ(Ω)$ of the quantity $h_Ω(E)/\log(1+d(E)/d(E,\partialΩ))$ over compact subsets $E$ of $Ω$ with at least two points, where $h_Ω(E)$ stands for the hyperbolic diameter of the set $E.$ We denote the upper half-plane by $\mathbb{H}$. Our main results claim that $κ(Ω)$ is positive if and only if the boundary of $Ω$ is uniformly perfect and that the inequality $κ(Ω)\leqκ(\mathbb{H})$ holds for all $Ω,$ where equality holds precisely when $Ω$ is convex.
In the present paper, we will study geometric properties of harmonic mappings whose analytic and co-analytic parts are (shifted) generated functions of completely monotone sequences.
For a given ring (domain) in $\overline{\mathbb{R}}^n$ 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 $n\ge 3\,,$ the standard definition of uniformly perfect sets in terms of Euclidean metric is equivalent to the boundedness of moduli of 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 $\mathbb{R}^n.$