Let S denote the class of functions f (z) = z + a 2 z 2 + ... 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.
In the present article we will examine the subclass of planar harmonic mappings. Let h(z) and g(z) are analytic functions in the open unit disc D = {z | |z| < 1} and having the power series represantation h(z) = z + a 2 z 2 +.. . and g(z) = b 1 z+b 2 z 2 +.. .. If f = h(z) + g(z) be the solution of the non-linear partial differential equation w q (z) = Dqg(z) Dqh(z) = fz fz with |w q (z)| < 1, h(z) q-convex function, then this class is called q-harmonic mappings for which analytic part is q-convex functions and the class of such functions is denoted by SHC(q), where D q h(z) = h(z)−h(qz) (1−q)z = f z , D q g(z) = g(z)−g(qz)
Let S * be the class of starlike functions and let SH be the class of harmonic mappings in the plane. In this paper we investigate harmonic mapping related to the starlike functions.
The aim of this paper is to give an investigation of the class of harmonic mappings related to the bounded boundary rotation. The class of bounded boundary rotation is generalized to the convex function.
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