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arXiv 1401.0259math.CV
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单向凸单叶调和映射的卷积性质

Convolution properties of univalent harmonic mappings convex in one direction

Raj Kumar, Sushma Gupta, Sukhjit Singh

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摘要：

设$\ast$和$\widetilde {\ast}$分别表示两个解析映射的卷积和一个解析映射与一个调和映射的卷积。Pokhrel [1]证明了：若$f = h+\overline{g}$是在$e^{iγ}$方向上凸的调和映射，$ϕ$是DCP类中的解析映射，则$f\widetilde{\ast} ϕ= h\widetilde{\ast}ϕ+ \overline{g\widetilde{\ast}ϕ}$也是在$e^{iγ}$方向上凸的，条件是$f\widetilde{\ast}ϕ$是局部单叶的且保意义的.本文给出了$f\widetilde{\ast} ϕ$是局部单叶保义的一般条件。一些有趣的应用的一般结果也被提出。

英文摘要：

Let $\ast$ and $\widetilde {\ast}$ denote the convolution of two analytic maps and that of an analytic map and a harmonic map respectively. Pokhrel [1] proved that if $f = h+\overline{g}$ is a harmonic map convex in the direction of $e^{iγ}$ and $ϕ$ is an analytic map in the class DCP, then $f\widetilde{\ast} ϕ= h\widetilde{\ast}ϕ+ \overline{g\widetilde{\ast}ϕ}$ is also convex in the direction of $e^{iγ}$, provided $f\widetilde{\ast}ϕ$ is locally univalent and sense-preserving. In the present paper we obtain a general condition under which $f\widetilde{\ast} ϕ$ is locally univalent and sense-preserving. Some interesting applications of the general result are also presented.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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