On the univalence of functions with logharmonic laplacian

In this paper, we prove Radó's theorem holds for functions of the form F(z)=r2L(z),L is logharmonic. We show that if F is of the form F(z)=r2L(z),|z|<1, where L(z)=h(z)g(z)¯ is logharmonic, then F is starlike iff ψ(z)=h(z)/g(z) is starlike. In addition, when F(z)=r2L(z)+H(z),|z|<1, where L is logharmonic and H is harmonic, we give the sufficient conditions for F to be locally univalent.