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Important Results on Janowski Starlike Log-harmonic Mappings Of Complex Order b

Melike AYDOGAN

Let H(D) be a linear space of all analytic functions defined on the open unit disc D. A sense-preserving log-harmonic function is the solution of the non-linear elliptic partial differential equation fz = w f f fz, where w(z) is analytic, satisfies the condition |w(z)| < 1 for every z ∈ D and is called the second dilatation of f. It has been shown that if f is a non-vanishing log-harmonic mapping then f can be represented by f(z) = h(z)g(z), where h(z) and g(z) are analytic in D with h(0) 6= 0, g(0) = 1([1]). If f vanishes at z = 0 but it is not identically zero, then f admits the representation f(z) = z |z| 2β h(z)g(z), where Reβ > − 1 2 , h(z) and g(z) are analytic in D with g(0) = 1 and h(0) 6= 0. The class of sense-preserving log-harmonic mappins is denoted by SLH. We say that f is a Janowski starlike log-harmonic mapping.If 1 + 1 b  zfz − zfz f − 1  = 1 + Aφ(z) 1 + Bφ(z) where φ(z) is Schwarz function. The class of Janowski starlike logharmonic mappings is denoted by S ∗ LH(A, B, b). We also note that, if (zh(z)) is a starlike function, then the Janowski starlike log-harmonic mappings will be called a perturbated Janowski starlike log-harmonic mappings. And the family of such mappings will be denoted by S ∗ P LH(A, B, b). The aim of this paper is to give some distortion theorems of the class S ∗ LH(A, B, b).