The Bohr radius and the Hadamard convolution operator

TL;DR

This paper introduces the concept of the Bohr radius for operator pairs, derives a general formula for Hadamard convolution operators, and applies it to differentiation and integration operators, extending existing theorems.

Contribution

It presents a new framework for the Bohr radius of operator pairs and generalizes a theorem on subordinate functions using this concept.

Findings

Derived a general formula for the Bohr radius of Hadamard convolution operators.

Applied the formula to differentiation and integration operators.

Extended B. Bhowmik and N. Das's theorem on subordinate functions.

Abstract

The concept of the Bohr radius of a pair of operators is introduced. In terms of the convolution function, a general formula for calculating the Bohr radius of the Hadamard convolution type operator with a fixed initial coefficient is obtained. We apply this formula to the problems of the Bohr radius of the operators of differentiation and integration. Using the concept of the Bohr radius of a pair of operators, we generalize the theorem of B.Bhowmik and N.Das on the comparison of majorant series of subordinate functions.