Bohr phenomenon for certain close-to-convex analytic functions

Abstract

We say that a class $\mathcal{B}$ of analytic functions $f$ of the form $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ satisfies a Bohr phenomenon if for the largest radius $R_f<1$, the following inequality $$\sum _{n=1}^{\infty }|a_n{z}^n|\le d(f(0),\partial f(\mathbb{D}))$$ holds for $|z|=r\le R_f$ and for all functions $f\in \mathcal{B}$. The largest radius $R_f$ is called Bohr radius for the class $\mathcal{B}$. In this article, we obtain Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes ${\mathcal{S}}_c^{\ast }(\varphi ),{\mathcal{C}}_c(\varphi ),{\mathcal{C}}_s^{\ast }(\varphi ),{\mathcal{K}}_s(\varphi )$. Using Bohr phenomenon for subordination classes \cite[Lemma 1]{bhowmik-2018}, we obtain some radius $R_f$ such that Bohr phenomenon for these classes holds for $|z|=r\le R_f$. Generally, in this case $R_f$ need not be sharp, but we show that under some additional conditions on $\varphi$, the radius $R_f$ becomes sharp bound. As a consequence of these results, we obtain several interesting corollaries on Bohr phenomenon for the aforesaid classes.

18 pages

References in corpus (1)

• Bohr phenomenon for locally univalent functions and logarithmic power series

Cited by in corpus (1)

• The Bohr inequality for certain harmonic mappings