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 vertical bar vertical bar z vertical bar < 1}, and let B be the set of all functions w(z) is an element of H(D) such that vertical bar w(z)vertical bar < 1 for all z is an element of D. A log-harmonic mapping is a solution of the non-linear elliptic partial differential equation (f) over bar ((z) over bar) = w(z) ((f) over bar /f) f(z), where w(z) is the second dilatation off and w(z) is an element of 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))over bar>, where H(z) and G(z) are in H(D), H(0) = G(0) = 1, and Re(R) > 0 for all z is an element of D. The class of such functions is denoted by P-LH. (C) 2012 Elsevier Inc. All rights reserved.