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 $\mathbf{D}$, we first obtain the starlike and convex radius for the linear combination of the differential operator $zf'$ of analytic functions $f$ in $\mathbf{D}.$ Then we obtain the starlike and convex radius for the differential operator $zf'$. Furthermore, we present the starlike and convex radius for the linear combination of the differential operator $zf'$ and analytic functions $f$ in $\mathbf{D}$. Our results imply the related result obtained by Gavrilov [Mat. Zametki, 7, 295–298 (1970)].