Rodrigo Hernández

Stable geometric properties of analytic and harmonic functions

Given any sense preserving harmonic mapping f=h+ḡ in the unit disk, we prove that for all |λ|=1 the functions f λ =h+λḡ are univalent ( resp . close-to-convex, starlike, or convex) if and only if the analytic functions F λ =h+λg are univalent ( resp . close-to-convex, starlike, or convex) for all such λ. We also obtain certain necessary geometric conditions on h in order that the functions f λ belong to the families mentioned above. In particular, we see that if f λ are univalent for all λ on the unit circle, then h is univalent.

On weighted compositions preserving the Carathéodory class

We characterize in various ways the weighted composition transformations which preserve the class

On convex combinations of convex harmonic mappings

\mathscr {P}

A Condition for Univalence in the Polydisk

P of normalized analytic functions in the disk with positive real part. We analyze the meaning of the criteria obtained for various special cases of symbols and identify the fixed points of such transformations.

Univalent harmonic mappings and linearly connected domains

Let f be a complex-valued harmonic mapping defined in the unit disk . We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which, roughly speaking, states that for analytic is Bloch if and only if is univalent.

Pre-Schwarzian and Schwarzian Derivatives of Harmonic Mappings

The family

Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian derivative

{\mathcal{F}}_{\unicode[STIX]{x1D706}}

QUASICONFORMAL EXTENSION OF HARMONIC MAPPINGS IN THE PLANE

of orientation-preserving harmonic functions