Multivalued Sakaguchi functions

Abstract

Let A be the class of functions f(z) of the form f(z) = z +a2z2 + ··· which are analytic in the open unit disc U = {z ∈ C||z| < 1}. In 1959 [5], K. Sakaguchi has considered the subclass of A consisting of those f(z) which satisfy Re zf (z) f(z)−f(−z)

> 0, where z ∈ U. We call such a functions “Sakaguchi Functions”. Various authors have investigated this class ([4], [5], [6]). Now we consider the class of functions of the form f(z) = zα(z +a2z2 +···+anzn +···) (0 <α< 1), that are analytic and multivalued in U, we denote the class of these functions by Aα, and we consider the subclass of Aα consisting of those f(z) which satisfy Re zDα z f(z) Dα z f(z)−Dα z f(−z)

> 0 (z ∈ U), where Dα z f(z) is the fractional derivative of order α of f(z). We call such a functions “Multivalued Sakaguchi Functions” and denote the class of those functions by Sα s . The aim of this paper is to investigate some properties of the class Sα s . 2000 Mathematical Subject Classification: Primary 30C45.