Sufficient conditions for univalence obtained by using the Ruscheweyh-Bernardi differential-integral operator

Abstract

"In this paper we introduce the Ruscheweyh-Bernardi differential-integral operator $T^m:A\to A$ defined by $$T^m[f](z)=(1-\lambda )R^m [f](z)+\lambda B^m[f](z),\ z\in U,$$ where $R^m$ is the Ruscheweyh differential operator (Definition \ref{d1.2}) and $B^m$ is the Bernardi integral operator (Definition \ref{d1.1}). By using the operator $T^m$, the class of univalent functions denoted by $T^m(\lambda ,\beta )$, $0\le \lambda \le 1$, $0\le \beta <1$, is defined and several differential subordinations are studied."