Subordination by convex functions

Abstract:

The following theorem is proven: Let $F(z)$ be convex and univalent in $\Delta = \{ z:|z| < 1\} ,F(0) = 1$. Let $f(z)$ be analytic in $\Delta ,f(0) = 1,f’(0) = \ldots = {f^{(n - 1)}}(0) = 0$, and let $f(z) \prec F(z)$ in $\Delta$. Then for all $\gamma \ne 0$, Re $\gamma \geqslant 0$, \[ {\gamma _z}^{ - \gamma }\int _0^z {{\tau ^{\gamma - 1}}f(\tau )d\tau \prec \gamma {z^{ - \gamma /n}}\int _0^{{z^{1/n}}} {{\tau ^{\gamma - 1}}F({\tau ^n})d\tau .} } \] This theorem, in combination with a method of D. Styer and D. Wright, leads to the following Corollary. Let $f(z),g(z)$ be convex univalent in $\Delta ,f(0) = f''(0) = g(0) = g''(0) = 0$. Then $f(z) + g(z)$ is starlike univalent in $\Delta$. Other applications of the theorem are concerned with the subordination of $f(z)/z$ where $f(z)$ belongs to certain classes of convex univalent functions.