Close-to-convex functions defined by fractional operator

Abstract

Let S denote the class of functions f(z) = z + a2z2+... analytic and univalent in the open unit disc D = {z ? C

z|<1}. Consider the subclass and S* of S, which are the classes ofconvex and starlike functions, respectively. In 1952, W. Kaplan introduced a class of analyticfunctions f(z), called close-to-convex functions, for which there exists?(Z) ? C, depending on f(z) with Re( f?(z)/??(z) ) > 0 in , and prove that every close-to-convex function is univalent. The normalized class of close-to-convex functions denoted by K. These classesare related by the proper inclusions C ? S* ? K ? S. In this paper, we generalize the close-to-convex functions and denote K(?) the class of such functions. Various properties of this class of functions is alos studied.