The Hardy class of a Bazilevič function and its derivative

Abstract:

The Bazilevič function $f(z)$ defined in $\Delta :|z| < 1$ by $f(z) \equiv {[\alpha \smallint _0^zP(\zeta )g{(\zeta )^\alpha }{\zeta ^{ - 1}}d\zeta ]^{1/\alpha }}$ where $g(\zeta )$ is starlike in $\Delta$, $P(\zeta )$ is regular with Re $P(\zeta ) > 0$ in $\Delta$ and $\alpha > 0$ is univalent. The class of such functions contains many of the special classes of univalent functions. The author determines the Hardy classes to which $f(z)$ and $f’(z)$ belong. In addition if $f(z) = \sum \nolimits _0^\infty {{a_n}{z^n}}$ the limiting value of $|{a_n}|/n$ is obtained.