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Toshiyuki Sugawa

Publications and source records attributed to Toshiyuki Sugawa.

At least 37 records · Page 2Linked to original sources
arXiv2019

Sharp inequalities for logarithmic coefficients and their applications

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 Lebedev-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 Lebedev-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 and then establish several consequences of them. 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.

math.CV↗
arXiv2019

Schur parameters and Carathéodory class

The Schur (resp. Carathéodory) class consists of all the analytic functions $f$ on the unit disk with $|f|\le 1$ (resp. $\Re f>0$ and $f(0)=1$). The Schur parameters $γ_0,γ_1,\dots (|γ_j|\le 1)$ are known to parametrize 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 $γ_1,\dots, γ_n$ with $|γ_j|\le 1.$ The mapping properties of those correspondences are also studied.

math.CV↗
arXiv2019

Geometric properties of the nonlinear resolvent of holomorphic generators

Let $f$ be the infinitesimal generator of a one-parameter semigroup $\left\{ F_{t}\right\} _{t\ge0}$ of holomorphic self-mappings of the open unit disk $Δ$. In this paper we study properties of the family $R$ of resolvents $(I+rf)^{-1}:Δ\toΔ~ (r\ge0)$ in the spirit of geometric function theory. We discovered, in particular, that $R$ forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of $R$ satisfies the Noshiro-Warschawski condition and is a starlike function of order at least $\frac12$,. This, in turn, implies that each element of $R$ is also a holomorphic generator. We mention also quasiconformal extension of an element of $R.$ Finally we study the existence of repelling fixed points of this family.

math.CV↗
arXiv2018

Characterizations for inner functions in certain function spaces

For $\frac12<p<\infty$, $0<q<\infty$ and a certain two-sided doubling weight $ω$, we characterize those inner functions $Θ$ for which $$\|Θ'\|_{A^{p,q}_ω}^q=\int_0^1 \left(\int_0^{2π} |Θ'(re^{iθ})|^p dθ\right)^{q/p} ω(r)\,dr<\infty.$$ Then we show a modified version of this result for $p\ge q$. Moreover, two additional characterizations for inner functions whose derivative belongs to the Bergman space $A_ω^{p,p}$ are given.

math.CV↗
arXiv2017

On the hyperbolic distance of $n$-times punctured spheres

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 quantitatively comparable with the systole of $X.$ We next propose a method to construct a distance function $d_X$ on a punctured sphere $X$ which is Lipschitz equivalent to the hyperbolic distance $h_X$ on $X.$ In particular, when the construction is based on a modified quasihyperbolic metric, $d_X$ is Lipschitz equivalent to $h_X$ with Lipschitz constant depending only on $Q(A).$

math.CV↗
arXiv2017

Spherical convexity and hyperbolic metric

Let $Ω$ be a domain in $\mathbb{C}$ with hyperbolic metric $λ_Ω(z)|dz|$ of Gaussian curvature $-4.$ Mejia and Minda proved in their 1990 paper that $Ω$ is (Euclidean) convex if and only if $d(z,\partialΩ)λ_Ω(z)\ge1/2$ for $z\inΩ,$ where $d(z,\partialΩ)$ denotes the Euclidean distance from $z$ to the boundary $\partialΩ.$ In the present note, we will provide similar characterizations of spherically convex domains in terms of the spherical density of the hyperbolic metric.

math.CV↗
arXiv2017

Geometric properties of the shifted hypergeometric functions

We will provide sufficient conditions for the shifted hypergeometric function $z_2F_1(a,b;c;z)$ 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.

math.CV↗
arXiv2017

A construction of trivial Beltrami coefficients

A measurable function $μ$ on the unit disk $\mathbb{D}$ of the complex plane with $\|μ\|_\infty<1$ is sometimes called a Beltrami coefficient. We say that $μ$ 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 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.

math.CV↗
arXiv2017

Construction of nearly hyperbolic distance on punctured spheres

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 comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute.

math.CV↗
arXiv2017

Hereditarily Non Uniformly Perfect Sets

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 plane of Hausdorff dimension 2 (and positive logarithmic capacity) which is hereditarily non uniformly perfect.

math.CV↗
arXiv2016

Spirallikeness of shifted hypergeometric functions

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 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.

math.CV↗
arXiv2016

A note on successive coefficients of convex functions

In this note, we investigate the supremum and the infimum of the functional $|a_{n+1}|-|a_{n}|$ for functions, convex and analytic on the unit disk, of the form $f(z)=z+a_2z^2+a_3z^3+\dots.$ We also consider the related problem to maximize the functional $|a_{n+1}-a_{n}|$ for convex functions $f$ with $f"(0)=p$ for a prescribed $p\in[0,2].$

math.CV↗
arXiv2015

On the second Hankel determinant of concave functions

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 coefficient body of order 2 for the class of analytic functions $φ(z)$ on $|z|<1$ with $|φ|<1$ and $φ(p)=p.$ We believe that this is helpful for other extremal problems concerning $a_2, a_3, a_4$ for normalized concave functions with a pole at $p.$

math.CV↗
arXiv2015

Analytic and geometric properties of open door functions

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 condition $|\Im[zf"(z)/f'(z)]|<c$ on $|z|<1$ implies starlikeness of an analytic function $f(z)$ on $|z|<1$ with $f(0)=f'(0)-1=0.$

math.CV↗
arXiv2015

Spherical density of hyperbolic metric and uniform perfectness

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.

math.CV↗
arXiv2015

Some extensions of the Open Door Lemma

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 sufficient condition for an analytic function on the unit disk to take its values in a given sector.

math.CV↗
arXiv2015

Quasiconformal extension of meromorphic functions with nonzero pole

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 $Σ_k(p)$. 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 $z=p\in(0,1)$ to belong to the class $Σ_k(p)$. Finally, we give a convolution property for functions in the class $Σ_k(p)$.

math.CV↗