Bohr Radius for Classes of Analytic Functions

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.