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 .

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Any harmonic function in the open unit disc D {z | |z| < 1} can be written as a sum of an analytic and antianalytic functions f h z g z , where h z and g z are analytic functions in D and are called the analytic part and the coanalytic part of f, respectively. Many important questions in the study of the classes of functions are related to bounds on the modulus of functions growth or the modulus of the derivative distortion . In this paper, we consider both of these questions.

Let A(p, n), n ≥ 1, p ≥ 1 be the class of all analytic functions in the open unit disc D = {z| |z| < 1} of the form f (z) = z p + a np+1 z np+1 + a np+2 z np+2 + • • •. Let g(z) be an element of A(p, n) such that g(z) satisfies the condition Re z g (z) g(z) > 0 for all z ∈ D, we then call g(z) the generalized p−valent starlike function in D. The class of such functions is denoted by S * (p, n). Let s(z) be an element of A(p, n) and g(z) be an element of S * (p, n). If the condition Re s(z) g(z) > 0 (z ∈ D), is satisfied then s(z) is called the generalized p−valent close-to-star function in D. The class of such functions is denoted by S * K(p, n). The aim of this paper is to give a distortion estimation and the radius of starlikeness of the class S * K(p, n).