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2 results

NOTE ON THE CONVOLUTION OF HARMONIC MAPPINGS
Part of
- Geometric function theory
LIULAN LI, SAMINATHAN PONNUSAMY
Journal:

Bulletin of the Australian Mathematical Society / Volume 99 / Issue 3 / June 2019

Published online by Cambridge University Press:

13 February 2019, pp. 421-431

Print publication:

June 2019
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Dorff et al. [‘Convolutions of harmonic convex mappings’, Complex Var. Elliptic Equ. 57(5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results.

Convex Subordination Chains in Several Complex Variables
Ian Graham, Hidetaka Hamada, Gabriela Kohr, John A. Pfaltzgraff
Journal:

Canadian Journal of Mathematics / Volume 61 / Issue 3 / 01 June 2009

Published online by Cambridge University Press:

20 November 2018, pp. 566-582

Print publication:

01 June 2009
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In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in ${{\mathbb{C}}^{n}}$. We also obtain a sufficient condition for injectivity of $f(z/\|z\|,\,\,\|z\|)$ on ${{B}^{n}}\backslash \{0\}$, where $f(z,t)$ is a convex subordination chain over $(0,1)$.