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

The classical Bohr theorem and its subsequent generalizations have become active areas of research, with investigations conducted in numerous function spaces. Let $\{{\psi }_n(r){\}}_{n=0}^{\mathrm{\infty }}$ be a sequence of non-negative continuous functions defined on $[0,1)$ such that the series ${\sum }_{n=0}^{\mathrm{\infty }}{\psi }_n(r)$ 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 $\text{D}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$. To achieve these, we employ the sequence of functions $\{{\psi }_n(r){\}}_{n=0}^{\mathrm{\infty }}$ in the majorant series rather than the conventional dependence on the basis sequence $\{r^n{\}}_{n=0}^{\mathrm{\infty }}$. 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 $\text{D}$. Moreover, we obtain a convolution counterpart of the Bohr theorem for harmonic mapping within the context of the Gaussian hypergeometric function