On the univalence of an integral operator

AbstractIn this work the author introduces a general integral operator and determines conditions for the univalence of this integral operator

Some generalizations on the univalence of an integral operator and quasiconformal extensions

By using the method of Loewner chains, we establish some sufficient conditions for the analyticity and univalency of functions defined by an integral operator. Also, we refine the result to a quasiconformal extension criterion with the help of Beckers's method.Comment: 12 pages, submitted to a journal for publication (19 April 2012

Prescribing the Preschwarzian in several complex variables

We solve the several complex variables preSchwarzian operator equation $[Df(z)]^{-1}D^2f(z)=A(z)$, z\in \C^n, where $A(z)$ is a bilinear operator and $f$ is a \C^n valued locally biholomorphic function on a domain in \C^n. Then one can define a several variables $f\to f_\alpha$ transform via the operator equation $[Df_\alpha(z)]^{-1}D^2f_\alpha(z)=\alpha[Df(z)]^{-1}D^2f(z)$, and thereby, study properties of $f_\alpha$. This is a natural generalization of the one variable operator $f_\alpha(z)$ in \cite{DSS66} and the study of its univalence properties, e.g., the work of Royster \cite{Ro65} and many others. M\"{o}bius invariance and the multivariables Schwarzian derivative operator of T. Oda \cite{O} play a central role in this work