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A general version of the Rado–Kneser–Choquet theorem implies that a piecewise constant sense-preserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.

By operating with the Scale Relativity Theory in the dynamics of complex systems, we can achieve a description of these complex systems through a holographic-type perspective. Then, gauge invariances of a Riccati-type become functional in complex system dynamics, which implies several consequences: conservation laws (in particular, for classical dynamics, the kinetic momentum conservation law), simultaneity and synchronization between the structural units (of a complex system) dynamics, and temporal patterns through harmonic mappings.

We study harmonic mappings of the form $f(z) = h(z) - \overline{z}$, where $h$ is an analytic function. In particular we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of $f$, where the Jacobian of $f$ is non-vanishing, it is known that this index has similar properties as the classical multiplicity of zeros of analytic functions. Little is known about the index of zeros on the critical set, where the Jacobian vanishes; such zeros are called singular zeros. Our main result is a characterization of the index of singular zeros, which enables one to determine the index directly from the power series of $h$., Oberwolfach Preprints;2017,03

Let S∗ be the class of starlike functions and let SH be the class of harmonic mappings in the plane. In this paper we investigate harmonic mapping related to the starlike functions. Publisher's Version

Let SH be the class of harmonic mappings defined by SH = { f = h z + g z | h z = z + ∑∞ n=2 anz n , g z = b1z + ∑∞ n=2 bnz n , b1 < 1 } where h z and g z are analytic. Additionally f z ∈ SH α ⇔ zh′ z − zg′ z h z + g z − 1 − b1 1 + b1 < 1 − b1 1 + b1 − α, z ∈ U, 0 6 α < 1 − b1 1 + b1 In the present work, by considering the analyticity of the functions defined by R. M. Robinson [7], we discuss the applications to the harmonic mappings.