New extension of Alexander and Libera integral operators

Abstract

Let T be the class of analytic functions in the open unit disc U with f(0) = 0 and f ′ (0) = 1. For f(z) ∈ T, the Alexander integral operator $A_{−1}$f(z), the Libera integral operator $L_{−1}$f(z) and the Bernardi integral operator $B_{−1}$f(z) were considered before. Using $A_{−1}$f(z) and $L_{−1}$1f(z), a new integral operator Fλf(z) is considered. After discuss some properties of dominant for Fλf(z), another new integral operator $O_{−1}$f(z) of f(z) ∈ T is discussed. The object of the present paper is to discuss the dominant of new integral operators Fλf(z) and $O_{−1}$f(z) concerning with some starlike functions and convex functions in U.