2003] were introduced a notion of Schwarzian derivative for a locally univalent harmonic mapping and showed that it retains some of the classical properties of the Schwarzian of an analytic function.

Harmonic mappings in the plane are univalent complex-valued harmonic functions whose real and imaginary parts are not necessarily conjugate.

It is known that every composition of a harmonic mapping with an analytic function is harmonic, but this useful fact does not always hold for p-harmonic mappings (p > 1).

A complex-valued harmonic mapping with positive Jacobian in D is known to satisfy the Beltrami equation of second kind [bar.

It is known that a harmonic mapping of an analytic function is harmonic, but an analytic function of a harmonic mapping is not necessarily harmonic (cf.

What is the harmonic mapping if all its pre-compositions or postcompositions by any harmonic mapping are still harmonic?

k] are complex-valued

harmonic mappings in D for all k [member of] {1, .

Among them are a simple numerical approach to the Riemann hypothesis, aunifying construction for measure-valued continuous and discrete branching processes, examples of quantitative universal approximation,

harmonic mappings with quadilateral image, meromorphic approximation on noncompact Riemann surfaces, a family of outer functions, the universality of series in Banach space, recent progress on fine differentiability and fine harmonicity, reversibility questions in groups arising from analysis, and the generalized binomial theorem.

Cohen); Birational aspects of the geometry of Mg(Gavril Farkas); The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces (Samuel Grushevsky and Igor Krichever); Brill-Noether theory (Joe Harris); GL+2(R)-orbit closures via topological splittings (Pascal Hubert, Erwan Lanneau, and Martin Moller);

Harmonic mappings and moduli spaces of Riemann surfaces (Jurgen Jost and Shing Tung Yau); Algebraic structures on the topology of moduli spaces of curves and maps (Y.

Silverman: Inequalities associating hypergeometric functions with planer

harmonic mappings, J.

Duran's topics include general properties of

harmonic mappings,

harmonic mappings into convex regions, harmonic self-mappings of the disk, harmonic univalent functions, external problems, mapping problems, minimal surfaces and curvature of minimal surfaces, with particular attention to the Weierstrass-Enneper representation.

For that purpose, let us first recall the definition of convolution of two

harmonic mappings.