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 of convex and starlike functions, respectively. In 1952, W. Kaplan introduced a class of analytic functions f(z), called close-to-convex functions, for which there exists φ(z) ∈ C, depending on f(z) with Re( f(z) φ(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 classes are related by the proper inclusions C ⊂ S∗ ⊂ K ⊂ S. In this paper, we generalize the close-to-convex functions and denote K(λ) the class of such functions. Various properties of this class of functions is alos studied.