Bounded log-harmonic functions with positive real part

Abstract

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 .

The Bohr radius for starlike logharmonic mappings

“…was recently investigated in [24]. If f is a non-constant logharmonic mapping of U which vanishes only at z = 0, then [25] f admits the representation…”

On geometrical properties of starlike logharmonic mappings

“…Note that f (0) = 0 if and only if m = 0, and that a univalent logharmonic mapping on U vanishes at the origin if and only if m = 1, that is, f has the form f (z) = z|z| 2β h(z)g(z), where Re(β) > −1/2 and 0 / ∈ (hg)(U ). This class has been studied extensively in recent years, for instance in [1,2,3,4,5,6,7,8,9,11,12,13,21,22,26] .…”

On rotationally starlike logharmonic mappings

“…Note that if and only if , and that a univalent logharmonic mapping on U vanishes at the origin if and only if , that is, f has the form where and This class has been studied extensively in recent years, for instance, in the works of –, and more recently in , , , .…”