Bohr radius for invariant families of bounded analytic functions and certain Integral transforms

TL;DR

This paper refines the Bohr radius for invariant families of bounded analytic functions and establishes Bohr inequalities for Fourier and Laplace transforms in specific domains, generalizing previous results and providing sharp estimates.

Contribution

It introduces a refined Bohr radius for invariant families and extends Bohr inequalities to certain integral transforms in new domains.

Findings

Refined Bohr radius for invariant families of bounded analytic functions.

Bohr inequalities established for Fourier and Laplace transforms in specified domains.

Sharp estimates obtained for the Laplace transform case.

Abstract

In this paper, we first obtain a refined Bohr radius for invariant families of bounded analytic functions on unit disk $ \mathbb{D} $. Then, we obtain Bohr inequality for certain integral transforms, namely Fourier (discrete) and Laplace (discrete) transforms of bounded analytic functions $ f(z)=\sum_{n=0}^{\infty}a_nz^n $, in a simply connected domain \begin{align*} \Omega_\gamma=\biggl\{z\in\mathbb{C}: \bigg|z+\dfrac{\gamma}{1-\gamma}\bigg|<\dfrac{1}{1-\gamma}\;\mbox{for}\; 0\leq \gamma<1\biggr\}, \end{align*} where $ \Omega_0=\mathbb{D} $. These results generalize some existing results. We also show that a better estimate can be obtained in radius and inequality can be shown sharp for Laplace transform of $ f $.