Extension Operators Preserving Janowski Classes of Univalent Functions

Abstract

In this paper, our main interest is devoted to study the extension op-, where α, β ≥ 0. We shall prove that if f ∈ S can be embedded as the first element of a g-Loewner chain with g : U → C given by g(ζ) = (1 + Aζ)/(1 + Bζ), |ζ| < 1, and −1 ≤ B < A ≤ 1, then F = Φ n,α,β (f ) can be embedded as the first element of a g-Loewner chain on the unit ballAs a consequence, the operator Φ n,α,β preserves the notions of Janowski starlikeness on B n and Janowski almost starlikeness on B n . Particular cases will be also mentioned.On the other hand, we are also concerned about some radius problems related to the operator Φ n,α,β and the Janowski class S * (a, b). We compute the radius S * (a, b) of the class S (respectively S * ).

Extension operators and Janowski starlikeness with complex coe cients

“…By making use of the same idea, we also prove that Φ n,Q conserves these classes when Q ≤ M (a, b), where M (a, b) is a constant depending on the parameters a and b. These results generalize the properties obtained in [13,14], for the Janowski classes with real parameters.…”