### Kernel convergence and biholomorphic mappings in several complex variables.

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We obtain an extension of Jack-Miller-Mocanu’s Lemma for holomorphic mappings defined in some Reinhardt domains in ${\u2102}^{n}$. Using this result we consider first and second order partial differential subordinations for holomorphic mappings defined on the Reinhardt domain ${B}_{2p}$ with p ≥ 1.

We define and investigate the notion of k-convexity in the sense of Mejia-Minda for domains in ℂⁿ and also that of k-convex mappings on the Euclidean unit ball.

Let f(z,t) be a Loewner chain on the Euclidean unit ball B in ℂⁿ. Assume that f(z) = f(z,0) is quasiconformal. We give a sufficient condition for f to extend to a quasiconformal homeomorphism of ${\mathbb{R}}^{2n}$ onto itself.

The authors obtain a generalization of Jack-Miller-Mocanu’s lemma and, using the technique of subordinations, deduce some properties of holomorphic mappings from the unit polydisc in ${\u2102}^{n}$ into ${\u2102}^{n}$.

In this paper we consider non-normalized univalent subordination chains and the connection with the Loewner differential equation on the unit ball in ${\u2102}^{n}$. To this end, we study the most general form of the initial value problem for the transition mapping, and prove the existence and uniqueness of solutions. Also we introduce the notion of generalized spirallikeness with respect to a measurable matrix-valued mapping, and investigate this notion from the point of view of non-normalized univalent subordination...

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