Abstract
Let A be the class of analytic functions in the open unit disk U. We define circle minus(alpha,b) : A -> A by (circle minus(alpha,beta) f) (z) := G(2 -a)z(a)D(Z)(a)(G(2 - b)z(b) D(Z)(b)f(z)), (a,b2,3,4 . . .),where D(Z)(y)fis the fractional derivative of f of order gamma. If alpha,beta is an element of[0,1], then a function f in A is said to be in the class SP alpha,beta if circle minus(alpha,beta) f is a parabolic starlike function. In this paper, several properties and characteristics of the class SP alpha,beta are investigated. These include subordination, characterization and inclusions, growth theorems, distortion theorems, and class-preserving operators. Furthermore, sandwich theorem related to the fractional derivative is proved. Copyright (C) 2009 O. Al-Refai and M. Darus.