In this paper, we show the existence of Landau constant for biharmonic mappings of the form F (z) = |z| 2 G(z) + K(z), |z| < 1, where G and K are harmonic.

We analyze the univalence of the solutions of the biharmonic equation. In particular, we show that if F is a biharmonic map in the form F(z) = r 2 G(z), |z| < 1, where G is harmonic, then F is starlike whenever G is starlike. In addition, when F(z) = r 2 G(z) + K(z), |z| < 1, where G and K are harmonic, we show that F is locally univalent whenever G is starlike and K is orientation preserving.

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Functions in H ■ //(D) are sense-preserving of the form f = h ■ g where h and g are in H(D). Such functions are solutions of an elliptic nonlinear P.D.E. that is studied in detail especially for its univalent solutions.

This paper surveys recent advances on univalent logharmonic mappings defined on a simply or multiply connected domain. Topics discussed include mapping theorems, logharmonic automorphisms, univalent logharmonic extensions onto the unit disc or the annulus, univalent logharmonic exterior mappings, and univalent logharmonic ring mappings. Logharmonic polynomials are also discussed, along with several important subclasses of logharmonic mappings.