Bounded log-harmonic functions with positive real part

Abstract

Let 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 equation fz¯¯=w(z)(f¯/f)fz, where w(z) is the 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 PLH.