Bounded log-harmonic functions with positive real part

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 f z ¯ ¯ = w ( z ) ( f ¯ / f ) f z , 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 R e ( R ) > 0 for all z ∈ D . The class of such functions is denoted by P L H .