Growth theorems for perturbated starlike log-harmonic mappings of complex order

Abstract

Let H(D) be the linear space of all analytic functions defined on the open unit disc D = {z ∈ C : |z| < 1}. A sense-preserving logharmonic mapping is the solution of the non-linear elliptic partial differential equation fz¯ = wfz ¡ f /f¢ , where w(z) ∈ H(D) is the second dilatation of f such that |w(z)| < 1 for every z ∈ D. It has been shown that if f is a non-vanishing log-harmonic mapping, then f can be expressed as f = h(z)g(z), where h(z) and g(z) are analytic in D. If f vanishes at z = 0 but it is not identically zero, then f admits the representation f = z|z| 2βh(z)g(z), where Reβ > −1/2, h(z) and g(z) are analytic in D, g(0) = 1, h(0) 6= 0 (see [1], [2], [3]). Let f = zh(z)g(z) be a univalent log-harmonic mapping. We say that f is a starlike log-harmonic mapping of complex order b (b 6= 0, complex) if Re ½ 1 + 1 b µ zfz − zf¯ z¯ f − 1 ¶¾ > 0, z ∈ D.

The class of all starlike log-harmonic mappings of complex order b is denoted by S ∗ LH(1 − b). We also note that if zh(z) is a starlike function of complex order b, then the starlike log-harmonic mapping f = zh(z)g(z) will be called a perturbated starlike log-harmonic mapping of complex order b, and the family of such mappings will be denoted by S ∗ LH(p)(1 − b). The aim of this paper is to obtain the growth theorems for the perturbated starlike log-harmonic mappings of complex order.