For p-valently spirallike and p-valently Robertson functions in the open unit disk U, reciprocal classes S p (α, β), and C p (α, β) are introduced. The object of the present paper is to discuss some interesting properties for functions f(z) belonging to the classes S p (α, β) and C p (α, β).

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.

For all q ∈ (0, 1) and 0 ≤ α < 1 we define a class of analytic functions, so-called q-starlike functions of order α on the open unit disc D = {z : |z| < 1} . We will study this class of functions and explore some inclusion properties with the well-known class Starlike functions of order α.

Quantum calculus had been used first time by M.E.H.Ismail, E.Merkes and D.Steyr in the theory of univalent functions [5].In this present paper we examine the subclass of univalent functions which is defined by quantum calculus.

a b s t r a c tLet H(D) be the linear space of all analytic functions defined on the open unit disc D = {z | |z| < 1}, and let B be the set of all functions w(z) ∈ H(D) such that |w(z)| < 1 for all z ∈ D. A log-harmonic mapping is a solution of the non-linear elliptic partial differential equationthe second dilatation of f and w(z) ∈ B. In the present paper we investigate the set of all log-harmonic mappings R defined on the unit disc D which are of the form R = H(z)G(z), where H(z) and G(z) are in H(D), H(0) = G(0) = 1, and Re(R) > 0 for all z ∈ D. The class of such functions is denoted by P LH .