Radius Problems for Some Subclasses of Analytic Functions

Abstract

In this work, we define the class ${\mathcal {M}}(\alpha )$ of normalized analytic functions which satisfy the following two-sided inequality:

$$\begin{aligned} 1+\frac{\alpha -\pi }{2\sin \alpha }< {{\mathfrak {R}}}{{\mathfrak {e}}}\left\{ \frac{zf'(z)}{f(z)}\right\}<1+ \frac{\alpha }{2\sin \alpha } \quad |z|<1, \end{aligned}$$

where $\pi /2\le \alpha <\pi$. We obtain a sufficient condition for functions to be in the class ${\mathcal {M}}(\alpha )$ and solve several radius problems related to other well-known function classes.