Sharp starlike and convex radius for the differential operator of analytic functions

UDC 517.5 Under given coefficient conditions for analytic functions f in the unit disk D , we first obtain the starlike and convex radius for the linear combination of the differential operator z f ' of analytic functions f in D . Then we obtain the starlike and convex radius for the differential operator z f ' . Furthermore, we present the starlike and convex radius for the linear combination of the differential operator z f ' and analytic functions f in D . Our results imply the related result obtained by Gavrilov [Mat. Zametki, <strong>7</strong>, 295–298 (1970)].