A generalization of the Bohr-Rogosinski sum

TL;DR

This paper explores generalizations of the Bohr-Rogosinski and classical Bohr sums for analytic functions in the unit disk, extending results to subordinate classes and highlighting key implications.

Contribution

It introduces a broad generalization of the Bohr-Rogosinski sum for subordinate analytic functions, expanding the scope of classical results.

Findings

Derived new bounds for the generalized Bohr-Rogosinski sum.

Extended classical Bohr sum results to subordinate function classes.

Identified implications for well-known analytic function properties.

Abstract

In this paper, we investigate the Bohr-Rogosinski sum and the classical Bohr sum for analytic functions defined on the unit disk in a general setting. In addition, we discuss a generalization of the Bohr-Rogosinski sum for a class of analytic functions subordinate to the univalent functions in the unit disk. Several well-known results are observed from the consequences of our main results.