On the coefficients of Bazilevič functions

Abstract:

Let $B(\alpha )$ denote the class of normalized $(f(0) = 0,f’(0) = 1)$, Bazilevič functions of type $\alpha$ defined in $\Delta :|z| < 1,{\text {i.e.}}f(z) = {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }}$ where $g(\zeta )$ is starlike in $\Delta ,P(\zeta )$ is regular with $\Re P(\zeta ) > 0$ in $\Delta$ and $\alpha > 0$. Let ${B_m}(\alpha )$ denote the subclass of $B(\alpha )$ which is m-fold symmetric $(f({e^{2\pi i/m}}z) = {e^{2\pi i/m}}f(z),m = 1,2, \cdots )$. Functions in $B(\alpha )$ have been shown to be univalent. The authors obtain sharp coefficient inequalities for functions in ${B_m}(1/N)$ where N is a positive integer. In addition an example of a Bazilevič function which is not close-to-convex is given.