An extremal problem for univalent functions
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.
math.CV↗
arXiv subjects
Toshiyuki Sugawa
Publications and source records attributed to Toshiyuki Sugawa.
Notes on convex functions of order $α$
Marx and Strohhäcker 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 $k_α(z)/z$ if $f$ is convex of order $α$ for $1/2\leα<1$ and conjectured that this is true also for $0<α<1/2.$ Here, $k_α$ is the standard extremal function in the class of normalized convex functions of order $α$ and $k_0(z)=z/(1-z).$ We prove the conjecture and study geometric properties of convex functions of order $α.$ In particular, we prove that $(f+g)/2$ is starlike whenever $f$ and $g$ both are convex of order $3/5.$
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Univalence criteria and analogs of the John constant
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 $δ_0$ and $δ_1$ such that the conditions $e^{-δ_0/2}<|p(z)|<e^{δ_0/2}$ and $|p(w)/p(z)|<e^{δ_1}$ for $|z|,|w|<1$ respectively imply univalence of $f$ on the unit disk.
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Variability regions of close-to-convex functions
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 $|z|<1.$ We will 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 $\log[zf'(z)/f(z)].$
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Univalent functions with half-integral coefficients
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 also observe geometric properties of the twelve functions.
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On univalence of the power deformation $z(f(z)/z)^c$
In this note, we mainly concern the set $U_f$ of $c\in\mathbb{C}$ such that the power deformation $z(f(z)/z)^c$ is univalent in the unit disk $|z|<1$ for a given analytic univalent function $f(z)=z+a_2z^2+\cdots$ in the unit disk. We will show that $U_f$ is a compact, polynomially convex subset of the complex plane $\C$ 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 $λ$-lemma for the holomorphic family $z(f(z)/z)^c$ of injections parametrized over the interior of $U_f.$ We also give necessary or sufficient conditions for $U_f$ to contain $0$ or $1$ as an interior point.
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On $μ$-conformal homeomorphisms and boundary correspondence
We study the boundary correspondence under $μ$-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.
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On a multi-point Schwarz-Pick lemma
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 functions on the unit disk.
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Sharp norm estimate of Schwarzian derivative for a class of convex functions
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 $α,~0<α<1,$ and for uniformly convex functions.
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Quasiconformal extension of strongly spirallike functions
We show that a strongly $λ$-spirallike function of order $α$ can be extended to a $\sin(πα/2)$-quasiconformal automorphism of the complex plane for $-π/2<λ<π/2$ and $0<α<1$ with $|λ|<πα/2.$ In order to prove it, we provide several geometric characterizations of a strongly $λ$-spirallike domain of order $α.$ We also give a concrete form of the mapping function of the standard strongly $λ$-spirallike domain $U_{λ,α}$ of order $α.$ A key tool of the present study is the notion of $λ$-argument, which was developed by Y. C. Kim and the author.
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On power deformations of univalent functions
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 surprisingly, we will see that the set is described by the variability region of the quantity $zf'(z)/f(z),~|z|<1,$ for the class in most cases which we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.
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Correspondence between spirallike functions and starlike functions
Let $λ$ be a real number with $-π/2<λ<π/2.$ In order to study $λ$-spirallike functions, it is natural to measure the angle according to $λ$-spirals. Thus we are led to the notion of $λ$-argument. This fits well the classical correspondence between $λ$-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 1971.
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Hardy spaces and unbounded quasidisks
We study the maximal number $0\le h\le+\infty$ for a given plane domain $Ω$ such that $f\in H^p$ 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.
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Geometric invariants associated with projective structures and univalence criteria
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.
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Invariant Schwarzian derivatives of higher order
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 invariant Schwarzians.
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Metrics with conical singularities on the sphere and sharp extensions of the theorems of Landau and Schottky
An explicit formula for the generalized hyperbolic metric on the thrice--punctured sphere $¶\backslash \{z_1, z_2, z_3\}$ with singularities of order $α_j \le 1$ 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 \cite{Pic1905} and Heins \cite{Hei62}, while explicit formulas for the cases $α_1=α_2=1$ were given earlier by Agard \cite{AG} and recently by Anderson, Sugawa, Vamanamurthy and Vuorinen \cite{A}. We also establish precise and explicit lower bounds for the generalized hyperbolic metric. This extends work of Hempel \cite{Hem79} and Minda \cite{Min87b}. As applications, sharp versions of Landau-- and Schottky--type theorems for meromorphic functions are obtained.
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Dual mean value problem for complex polynomials
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.
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Some counterexamples in dynamics of rational semigroups
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.
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