The starlike radius for classes of regular bounded functions

Abstract:

Let ${B_0}(a)$ be the class of all functions $f$ defined on $|z| < 1$ such that (i) $f(z)$ is regular, (ii) $|f(z)| < 1$ (iii) $f(0) = 0$ (iv) $0 < |f’(0)| = a \leqslant 1$. For fixed $R,a \leqslant R < 1$, let ${B_0}(a;R)$ be that subclass having nonzero zeros at $z = {z_k},k = 1,2, \ldots$, such that $\prod |{z_k}| = R$. The subclass having no nonzero zeros is designated as ${B_0}(a;1)$. A sharp lower bound for $\operatorname {Re} [zf’(z)/f(z)]$ for the class ${B_0}(a;R),a \leqslant R \leqslant 1$, is obtained, and the radius of starlikeness is found. A covering theorem for the class is also obtained.