on some properties of log-harmonic functions product

In this paper we define a new subclass slh(k, γ; φ) of log-harmonic mappings, and then basic properties such as dilations, convexity on one direction and convexity of log functions of convex- exponent product of elements of that class are discussed. also we find sufficient conditions on β such that f ∈ slh(k, γ; φ) leads to f(z) = f(z)|f(z)|^2β ∈ slh(k, γ, φ). our results generalize the analogues of the earlier works in the combinations of harmonic functions.