Marx-Strohhäcker differential subordination systems

Abstract:

Let $f(z) = z + {a_2}{z^2} + \cdots$ be analytic in the unit disc $U$ and let $k(z) = z/(1 - z)$. The classic Marx-Strohhäcker result, that a convex (univalent) function $f$ is starlike of order $\frac {1}{2}$, can be written in terms of differential subordinations as \[ zf''(z)/f’(z) \prec zk''(z)/k’(z) \Rightarrow zf’(z)/f(z) \prec zk’(z)/k(z).\] The authors determine general conditions on $k$ for which this relation holds. They also determine a different set of general conditions on $k$ for which \[ zf’(z)/f(z) \prec zk’(z)/k(z) \Rightarrow f(z)/z \prec k(z)/z.\] Finally, differential subordinations with starlike superordinate functions are considered.