Generalized Bohr inequalities for certain classes of functions and their applications

Let $\mathcal{B}:=\{f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^{n}with\mathrm{\mid }f(z)\mathrm{\mid }<1forallz\in \mathbb{D}\}$. The improved version of the classical Bohr’s inequality \cite{Bohr-1914} states that if $f\in \mathcal{B}$, then the associated majorant series $M_f(r):={\sum }_{n=0}^{\mathrm{\infty }}\mathrm{\mid }{a}_n\mathrm{\mid }{r}^n\le 1$ holds for $\mathrm{\mid }z\mathrm{\mid }=r\le 1\mathrm{/}3$ and the constant $1\mathrm{/}3$ 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 $\{{\phi }_n(r){\}}_{n=0}^{\mathrm{\infty }}$ of non-negative continuous functions on $[0,1)$ in the place of $\{r^n{\}}_{n=0}^{\mathrm{\infty }}$ of the majorant series $M_f(r)$ introducing a weighted sequence of non-negative continuous functions $\{{\mathrm{\Phi }}_n(r){\}}_{n=0}^{\mathrm{\infty }}$ 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.