Bohr Phenomenon for Certain Close-to-Convex Analytic Functions

Abstract

We say that a class \({\mathcal {G}}\) 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 $$\begin{aligned} \sum \limits _{n=1}^{\infty } |a_{n}z^{n}| \le d(f(0),\partial f({\mathbb {D}}) ) \end{aligned}$$holds for \(|z|=r\le R_{f}\) and for all functions \(f \in {\mathcal {G}}\). The largest radius \(R_{f}\) is called Bohr radius for the class \({\mathcal {G}}\). In this article, we obtain the Bohr radius for certain subclasses of close-to-convex analytic functions. We establish the Bohr phenomenon for certain analytic classes \({\mathcal {S}}_{c}^{*}(\phi ),\,{\mathcal {C}}_{c}(\phi ),\, {\mathcal {C}}_{s}^{*}(\phi ),\, {\mathcal {K}}_{s}(\phi )\) and obtain the radius \(R_{f}\) such that the Bohr phenomenon for these classes holds for \(|z|=r\le R_{f}\). As a consequence of these results, we obtain several interesting corollaries about the Bohr phenomenon for the aforesaid classes.