Generalized Bohr inequalities for certain classes of functions and their applications

Abstract

Let $ \mathcal{B}:=\{f(z)=\sum_{n=0}^{\infty}a_nz^n\; \mbox{with}\; |f(z)|<1\;\mbox{for all}\; z\in\mathbb{D}\} $. The improved version of the classical Bohr's inequality \cite{Bohr-1914} states that if $f\in \mathcal{B}$, then the associated majorant series $M_f(r):={\sum }_{n=0}^{\infty }|a_n|r^n\le 1$ holds for $|z|=r\le 1/3$ and the constant $1/3$ cannot be improved. Bohr's original theorem and its subsequent generalizations remain active fields of study, driving investigations in a wide range of function spaces. In this paper, first we establish a generalized Bohr inequality for the class $\mathcal{B}$ by allowing a sequence $\{{\phi }_n(r){\}}_{n=0}^{\infty }$ of non-negative continuous functions on $[0,1)$ in the place of $\{r^n{\}}_{n=0}^{\infty }$ of the majorant series $M_f(r)$ introducing a weighted sequence of non-negative continuous functions $\{{\Phi }_n(r){\}}_{n=0}^{\infty }$ on $[0,1)$. Secondly, as a generalization, we obtain a refined version of the Bohr inequality for a certain class ${\tilde{G}}_{\mathcal{H}}^0(\beta )$ of harmonic mappings. All the results are proved to be sharp.

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