Bohr Radius for Classes of Analytic Functions

Abstract

A class $${\mathcal {M}}$$ consisting of analytic functions $$f(z)=\sum _{n=0}^{\infty }a_nz^n$$ in the unit disc $$\mathbb {D}$$ satisfies a Bohr phenomenon if there exists an $$r^* > 0$$ such that $$\begin{aligned} \sum _{n=1}^{\infty }|a_nz^n|\le d(f(0), \partial f(\mathbb {D})) \end{aligned}$$for every function $$f\in {\mathcal {M}},$$ and $$|z|=r \le r^*$$. The largest $$r^*$$ is the Bohr radius for the class $${\mathcal {M}}.$$ Here d is the Euclidean distance. In this paper, the Bohr radii are obtained when $${\mathcal {M}}$$ is the class consisting of convex univalent functions of order $$\alpha $$, $${\mathcal {M}}$$ a subclass of close-to-convex functions, as well as $${\mathcal {M}}$$ a subclass of functions with positive real part. An improved Bohr radius is obtained when the class treated have negative coefficients.