arXiv subjects
Publications and source records attributed to Hiroshi Yanagihara.
In this paper, we investigate several Bohr radii associated with the Cesáro operator, Bernardi integral operator, $β$-Cesáro operator, and discrete Fourier transform, all defined on a set of holomorphic mappings from the unit ball of a complex Banach space into the closure of the unit polydisc $\mathbb{D}^n$ within the space $\mathbb{C}^n$.
We study Loewner chains in $\mathcal{H}_0(\mathbb{D})$ without assuming univalence of each element. We prove a decomposition: every chain admits a factorization $f_t=F\circ g_t$, where $F$ is analytic on $\mathbb{D}(0,r)$ with $r=\lim_{t \nearrow \sup I} f_t'(0)$, and $\{g_t\}$ is a classical Loewner chain of univalent functions. Under a mild regularity assumption on $t \mapsto f_t'(0)$, we derive a partial differential equation that generalizes the Loewner--Kufarev equation. We then develop a Loewner theory for chains of universal covering maps. We characterize such chains in terms of domain families $\{Ω_t\}$: continuity and monotonicity of $\{f_t\}$ are equivalent to kernel continuity and monotonicity of $\{Ω_t\}$. We further show that the connectivity $C(Ω_t)=\#(\hat{\mathbb{C}}\setminus Ω_t)$ is a left-continuous nondecreasing function of $t$. Building on these results, we formulate a Loewner theory on Fuchsian groups and obtain evolution equations for deck transformations. As an application, we study hyperbolic metrics and establish a formula for the logarithmic derivative of the hyperbolic density along the chain. Our results provide a unified framework linking classical Loewner theory, covering maps, and the geometry of hyperbolic domains.
The main aim of this paper is to establish several Landau-type theorems for certain bounded poly-analytic functions and reduced poly-analytic functions that generalize some previously established results.
Recently in relation to the theory of non-commutative probability, a notion of evolution families $\{ω_{s,t}\}_{s \le t}$ is generalized that are only continuous in parameters, namely $(s,t) \mapsto ω_{s,t}$ is continuous with respect to locally uniform convergence on a planar domain. In this article we present various equivalence conditions to the continuous evolution families concerned with the left and right parameters. We also provide an example of a discontinuous evolution family in the last section.
We continue our study on variability regions in \cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region of variability $V_Ω^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g({\mathbb D}) \subset Ω, \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$ for each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 , \ldots , c_n) \in \mathbb{C}^{n+1}$, when $Ω\subsetneq\mathbb{C}$ is a convex domain, and $P$ is a conformal map of the unit disk ${\mathbb D}$ onto $Ω$. In the present article, we first show that in the case $n=0$, $j=-1$ and $c=0$, the result obtained in \cite{Ali-Vasudevarao-Yanagihara-2018} still holds when one assumes only that $Ω$ is starlike with respect to $P(0)$. Let $\mathcal{CV}(Ω)$ be the class of analytic functions $f$ in ${\mathbb D}$ with $f(0)=f'(0)-1=0$ satisfying $1+zf''(z)/f'(z) \in Ω$. As applications we determine variability regions of $\log f'(z_0)$ when $f$ ranges over $\mathcal{CV}(Ω)$ with or without the conditions $f''(0)= λ$ and $f'''(0)= μ$. Here $λ$ and $μ$ are arbitrarily preassigned values. By choosing particular $Ω$, we obtain the precise variability regions of $\log f'(z_0)$ for other well-known subclasses of analytic and univalent functions.
For $0<λ\le 1$, let $\mathcal{U}(λ)$ be the class analytic functions $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ in the unit disk $\mathbb{D}$ satisfying $|f'(z)(z/f(z))^2-1|<λ$ and $\mathcal{U}:=\mathcal{U}(1)$. In the present article, we prove that the class $\mathcal{U}$ is contained in the closed convex hull of the class of starlike functions and using this fact, we solve some extremal problems such as integral mean problem and arc length problem for functions in $\mathcal{U}$. By means of the so-called theory of star functions, we also solve the integral mean problem for functions in $\mathcal{U}(λ)$. We also obtain the estimate of the Fekete-Szegö functional and the pre-Schwarzian norm of certain nonlinear integral transform of functions in $\mathcal{U}(λ)$. Further, for the class of meromorphic functions which are defined in $Δ:=\{ζ\in\mathbb{\widehat{C}}:|ζ|>1\}$ and associated with the class $\mathcal{U}(λ)$, we obtain a sufficient condition for a function $g$ to be an extreme point of this class.
Let $z_0$ and $w_0$ be given points in the open unit disk $\mathbb{D}$ with $|w_0| < |z_0|$. Let $\mathcal{H}_0$ be the class of all analytic self-maps $f$ of $\mathbb{D}$ normalized by $f(0)=0$, and $\mathcal{H}_0 (z_0,w_0) = \{ f \in \mathcal{H}_0 : f(z_0) =w_0\}$. In this paper, we explicitly determine the variability region of $f''(z_0)$ when $f$ ranges over $\mathcal{H}_0 (z_0,w_0)$. We also show a geometric view of our main result by Mathematica.
Let $Ω$ be a convex domain in the complex plane ${\mathbb C}$ with $Ω\not= {\mathbb C}$, and $P$ be a conformal map of the unit disk ${\mathbb D}$ onto $Ω$. Let ${\mathcal F}_Ω$ be the class of analytic functions $g$ in ${\mathbb D}$ with $g({\mathbb D}) \subset Ω$, and $H_1^\infty ({\mathbb D})$ be the closed unit ball of the Banach space $H^\infty ({\mathbb D})$ of bounded analytic functions $ω$ in ${\mathbb D}$, with norm $\| ω\|_\infty = \sup_{z \in {\mathbb D}} |ω(z)|$. Let ${\mathcal C}(n) = \{ (c_0,c_1 , \ldots , c_n ) \in {\mathbb C}^{n+1}: \text{there exists} \; ω\in H_1^\infty ({\mathbb D}) \; \text{satisfying} \; ω(z) = c_0+c_1z + \cdots + c_n z^n + \cdots$ for ${z\in \mathbb D}\}$. For each fixed $z_0 \in {\mathbb D}$, $j=-1,0,1,2, \ldots$ and $c = (c_0, c_1 , \ldots , c_n) \in {\mathcal C}(n)$, we use the Schur algorithm to determine the region of variability $V_Ω^j (z_0, c ) = \{ \int_0^{z_0} z^{j}(g(z)-g(0))\, d z : g \in {\mathcal F}_Ω\; \text{with} \; (P^{-1} \circ g) (z) = c_0 +c_1z + \cdots + c_n z^n + \cdots \}$. We also show that for $z_0 \in {\mathbb D} \backslash \{ 0 \}$ and $c \in \textrm{Int} \, {\mathcal C}(n) $, $V_Ω^j (z_0, c )$ is a convex closed Jordan domain, which we determine by giving a parametric representation of the boundary curve $\partial V_Ω^j (z_0, c )$.
We study the class ${\mathcal C}(Ω)$ of univalent analytic functions $f$ in the unit disk $\mathbb{D} = \{z \in \mathbb{C} :\,|z|<1 \}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ satisfying \[ 1+\frac{zf"(z)}{f'(z)} \in Ω, \quad z\in \mathbb{D}, \] where $Ω$ will be a proper subdomain of ${\mathbb C}$ which is starlike with respect to $1 (\in Ω)$. Let $ϕ_Ω$ be the unique conformal mapping of ${\mathbb D}$ onto $Ω$ with $ϕ_Ω(0)=1$ and $ϕ_Ω'(0) > 0$ and $ k_Ω(z) = \int_0^z \exp \left(\int_0^t ζ^{-1} (ϕ_Ω(ζ) -1) \, d ζ\right) \, dt$. Let $L_r(f)$ denote the arclength of the image of the circle $\{z \in \mathbb{C} : \, |z|=r\}$, $r\in (0,1)$. The first result in this paper is an inequality $L_r(f) \leq L_r(k_Ω)$ for $f \in \mathcal{C} (Ω)$, which solves the general extremal problem $\max_{f \in {\mathcal C}(Ω)} L_r(f)$, and contains many other well-known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class ${\mathcal C}(Ω)$.