Generalized Bohr inequalities for K-quasiconformal harmonic mappings and their applications

TL;DR

This paper extends classical Bohr inequalities to K-quasiconformal harmonic mappings using a sequence of functions for sharper bounds, with applications including improved inequalities and a convolution version involving hypergeometric functions.

Contribution

It introduces generalized Bohr inequalities for K-quasiconformal harmonic mappings using a novel sequence-based approach, refining existing results and deriving new convolution inequalities.

Findings

Established sharp generalized Bohr inequalities for K-quasiconformal harmonic mappings.

Derived improved and refined Bohr inequalities for harmonic mappings.

Obtained a convolution version of the Bohr theorem involving hypergeometric functions.

Abstract

The classical Bohr theorem and its subsequent generalizations have become active areas of research, with investigations conducted in numerous function spaces. Let $\{\psi_n(r)\}_{n=0}^\infty$ be a sequence of non-negative continuous functions defined on $[0,1)$ such that the series $\sum_{n=0}^\infty \psi_n(r)$ converges locally uniformly on the interval $[0, 1)$. The main objective of this paper is to establish several sharp versions of generalized Bohr inequalities for the class of $K$-quasiconformal sense-preserving harmonic mappings on the unit disk $\D := \{z \in \mathbb{C} : |z| < 1\}$. To achieve these, we employ the sequence of functions $\{\psi_n(r)\}_{n=0}^\infty$ in the majorant series rather than the conventional dependence on the basis sequence $\{r^n\}_{n=0}^\infty$. As applications, we derive a number of previously published results as well as a number of sharply improved and refined Bohr inequalities for harmonic mappings in $\D$. Moreover, we obtain a convolution counterpart of the Bohr theorem for harmonic mapping within the context of the Gaussian hypergeometric function