On a problem of Gronwall for Bazilevič functions

Abstract:

Let $B(\alpha ,\beta ),\alpha$ positive, $\beta$ real, denote the class of normalized univalent Bazilevič functions in $K = \{ z:|z| < 1\}$ of type $\alpha ,\beta$. Let $B = { \cup _{\alpha ,\beta }}B(\alpha ,\beta )$. Let $\alpha ,0 \leq \alpha \leq 2$, and $\alpha ,0 < \alpha < \infty$, be fixed and suppose that $f(z) = z + a{z^2} + \cdots$ is in $B(\alpha ,0)$. In this paper for given ${z_0} \in K$, the author finds a sharp upper bound for $|f({z_0})|$. Also, a sharp asymptotic bound is obtained for ${(1 - r)^2}{\max _{|z| = r}}|f(z)|$. Finally, a sharp asymptotic bound is found for ${(1 - r)^2}{\max _{|z| = r}}|f(z)|$ when f is in B with second coefficient a.