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

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