Extensions of Some Known Differential and Integral Operators

Abstract

AbstractIn this paper, the operator \(L^{m}:A\rightarrow A\) is introduced by \(L^{m}[f](z)=(1-\lambda )R^{m}[f](z)+\lambda I^{m}[f](z)\), \(z\in U\), a differential–integral operator where \(R^{m}\) is Ruscheweyh differential operator and \(I^{m}\) is Alexander integral operator. By using the operator \(L^{m}\) the class denoted by \(M^{m}(\lambda ,\beta )\), \(0\le \lambda \le 1\) , \(0\le \beta <1\) is defined. Using the operator \(L^{m}\), several differential subordinations are studied and from this study we obtain a condition for univalence of functions from class A.KeywordsAnalytic functionDifferential operatorIntegral operatorConvex functionThe best dominantDifferential subordinationUnivalent functionClassification30C2030C4530A10