# On the Bohr inequality with a fixed zero coefficient

## TL;DR

This paper extends the classical Bohr inequality to quasi-subordination families, improving bounds for bounded analytic functions and harmonic mappings, and determines the Bohr radius for odd analytic functions.

## Contribution

It introduces the Bohr phenomenon for quasi-subordinate functions and establishes exact bounds, enhancing classical results for various function classes.

## Key findings

- Classical Bohr's inequality is established for quasisubordinate functions.
- Improved bounds for bounded analytic functions and harmonic mappings are obtained.
- The Bohr radius for odd analytic functions is determined.

## Abstract

In this paper, we introduce the study of the Bohr phenomenon for a quasi-subordination family of functions, and establish the classical Bohr's inequality for the class of quasisubordinate functions. As a consequence, we improve and obtain the exact version of the classical Bohr's inequality for bounded analytic functions and also for $K$-quasiconformal harmonic mappings by replacing the constant term by the absolute value of the analytic part of the given function. We also obtain the Bohr radius for the subordination family of odd analytic functions.