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

Zhenyong Hu; H. M. Srivastava; Ying Zhang

doi:10.3842/umzh.v76i10.7788

Zhenyong Hu School of Mathematics and Statistics, Zhoukou Normal University, China
H. M. Srivastava Department of Mathematics and Statistics, University of Victoria, British Columbia, Canada; Center for Converging Humanities, Kyung Hee University, Republic of Korea; Department of Mathematics and Informatics, Azerbaijan University, Baku, Azerbaijan, and Section of Mathematics, International Telematic University Uninettuno, Rome, Italy
Ying Zhang School of Mathematics and Statistics, Xinyang College, China

DOI: https://doi.org/10.3842/umzh.v76i10.7788

Keywords: analytic functions; starlike functions; convex functions; differential operator.

Abstract

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)].

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