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Shankey Kumar

Publications and source records attributed to Shankey Kumar.

9 recordsLinked to original sources
arXiv2026

On the Boundary Schwarz lemma and the rigidity theorem for certain mappings

In this article, we characterize the holomorphic mappings from $B_{\ell_p^n}\times\mathbb{D}^{m}$ into $\mathbb{D}^{m}$ for $p\in \{2,\infty\}$. In addition, we give a simple proof for the boundary Schwarz lemma for vector valued holomorphic functions, which also extends the existing result. Also, we obtain the boundary Schwarz lemma for pluriharmonic self-mappings of the unit ball $B_{\ell_p^n}$, $p \in [2,\infty]$. Furthermore, we establish the boundary rigidity theorem for holomorphic self-mappings of $B_{\ell_p^n}$, $p \in (1,\infty)$.

math.CV↗
arXiv2025

Schwarz-Pick type lemma and Landau type theorem for $\alpha$-harmonic mappings

The aim of this paper is twofold. First, we obtain a Schwarz-Pick type lemma for the $\alpha$-harmonic mapping $u=P_{\alpha}[\phi]$, where $\phi\in L^{p}(\mathbb{S}^{n-1},\mathbb{R} )$ and $p\in[1,\infty]$. We get an explicit form of the sharp function $\mathbf{C}_{\alpha, q}(x)$ in the inequality $|\nabla u(x)| \leq \mathbf{C}_{\alpha, q}(x)\|\phi\|_{L^p(\mathbb{S}^{n-1}, \mathbb{ R} )}$. Second, we prove a Landau type theorem for $u=P_{\alpha}[\phi]$, where $\phi\in L^{\infty}(\mathbb{S}^{n-1},\mathbb{R}^{n})$. These results generalize and extend the corresponding results due to Kalaj (Complex Anal. Oper. Theory, 2024) and Khalfallah et al. (Mediterr. J. Math., 2021).

math.AP↗
arXiv2023

Multi-dimensional Bohr radii of Banach space valued holomorphic functions

In this article, we study the multi-dimensional Bohr radii of holomorphic functions defined on the Banach sequence spaces with values in the Banach spaces. For the case of finite dimensional Banach spaces, we exhibit the exact asymptotic growth of the Bohr radius. To achieve our goal in the finite case, we use $\ell_{p'}$-summability of certain coefficients of a given polynomial in terms of its uniform norm on $\ell_p^n$. The infinite case is handled using the techniques developed in recent years from the work of Defant, Maestre and Schwarting. We crucially use several properties of the symmetric $M$-linear mapping associated with a homogeneous polynomial of degree $M$ in our analysis. Furthermore, we study the bounds of the arithmetic Bohr radius of Banach space-valued holomorphic functions defined on the Banach sequence spaces, which generalises the work of Defant, Maestre, and C. Prengel in this direction.

math.FA↗
arXiv2022

Asymptotic value of the multidimensional Bohr radius

This article determines the exact asymptotic value of the Bohr radii and the arithmetic Bohr radii for the holomorphic functions defined on the unit ball of the $\ell_p^n$ space and having values in the simply connected domain of $\mathbb{C}$. Moreover, we investigate sharp Bohr radius for four distinct categories of holomorphic functions. These functions map the bounded balanced domain $G$ of a complex Banach space $X$ into the following domains: the right half-plane, the slit domain, the punctured unit disk, and the exterior of the closed unit disk.

math.CV↗
arXiv2021

A generalization of the Bohr inequality for bounded analytic functions on simply connected domains and its applications

Bohr's classical theorem and its generalizations are now active areas of research and have been the source of investigations in numerous function spaces. In this article, we study a generalized Bohr's inequality for the class of bounded analytic functions defined on the simply connected domain $$ \Omega_{\gamma}:=\bigg\{z\in \mathbb{C}: \bigg|z+\frac{\gamma}{1-\gamma}\bigg|<\frac{1}{1-\gamma}\bigg\}, \,\ \text{for } 0\leq \gamma<1. $$ Part of its applications, we calculate the Bohr-type radii for some known integral operators.

math.CV↗
arXiv2018

Properties of $\beta$-Ces\`aro operators on $\alpha$-Bloch space

For each $ \alpha > 0 $, the $\alpha$-Bloch space is consisting of all analytic functions $f$ on the unit disk satisfying $ \sup_{|z|<1} (1-|z|^2)^\alpha |f'(z)| < + \infty.$ In this paper, we consider the following complex integral operator, namely the $\beta$-Ces\`{a}ro operator \begin{equation} C_\beta(f)(z)=\int_{0}^{z}\frac{f(w)}{w(1-w)^{\beta}}dw \nonumber \end{equation} and its generalization, acting from the $\alpha$-Bloch space to itself, where $f(0)=0$ and $\beta\in\mathbb{R}$. We investigate the boundedness and compactness of the $\beta$-Ces\`{a}ro operators and their generalization. Also we calculate the essential norm and spectrum of these operators.

math.FA↗