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Vasudevarao Allu

Publications and source records attributed to Vasudevarao Allu.

75 records · Page 5Linked to original sources

arXiv2020

Bohr phenomenon for certain Subclasses of Harmonic Mappings

The Bohr phenomenon for analytic functions of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r_{f}$, $0<r_{f}<1$, such that the inequality $\sum_{n=0}^{\infty} |a_{n}z^{n}| \leq 1$ holds whenever the inequality $|f(z)|\leq 1 $ holds in the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$. The exact value of this largest radius known as Bohr radius, which has been established to be $r_{f}=1/3$. The Bohr phenomenon \cite{Abu-2010} for harmonic functions $f$ of the form $f(z)=h(z)+\overline {g(z)}$, where $h(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ and $g(z)=\sum_{n=1}^{\infty} b_{n}z^{n}$ is to find the largest radius $r_{f}$, $0<r_{f}<1$ such that $$\sum\limits_{n=1}^{\infty} (|a_{n}|+|b_{n}|) |z|^{n}\leq d(f(0),\partial f(\mathbb{D})) %\quad\mbox { for } |z|\leq r_{f}. $$ holds for $|z|\leq r_{f}$, here $d(f(0),\partial f(\mathbb{D})) $ denotes the Euclidean distance between $f(0)$ and the boundary of $f(\mathbb{D})$. In this paper, we investigate the Bohr radius for several classes of harmonic functions in the unit disk $\mathbb{D}.$

math.CV↗

arXiv2020

Zalcman Conjecture for certain analytic and univalent functions

Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}$ of the form $f(z)= z+\sum_{n=2}^{\infty}a_n z^n$ and $\mathcal{S}$ denote the class of functions $f\in\mathcal{A}$ which are univalent ({\it i.e.}, one-to-one). In 1960s, L. Zalcman conjectured that $|a_n^2-a_{2n-1}|\le (n-1)^2$ for $n\ge 2$, which implies the famous Bieberbach conjecture $|a_n|\le n$ for $n\ge 2$. For $f\in \mathcal{S}$, Ma \cite{Ma-1999} proposed a generalized Zalcman conjecture $$|a_{n}a_{m}-a_{n+m-1}|\le (n-1)(m-1) $$ for $n\ge 2, m\ge 2$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying $$ \left|f'(z)\left(\frac{z}{f(z)}\right)^2-1 \right|< 1 \quad\mbox{ for } z\in\mathbb{D}. $$ and $\mathcal{F}$ denote the class of functions $f\in \mathcal{A}$ satisfying ${\rm Re\,}(1-z)^{2}f'(z)>0$ in $\mathbb{D}$. In the present paper, we prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{U}$ using extreme point theory. We also prove the Zalcman conjecture and generalized Zalcman conjecture for the class $\mathcal{F}$ for the initial coefficients.

math.CV↗

arXiv2019

An application of the Schur algorithm to variability regions of certain analytic functions

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 $\partial V_Ω^j (z_0, c )$.

math.CV↗