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arXiv · 1703.03595

Convolution of a harmonic mapping with $n$-starlike mappings and its partial sums

Subzar Beig·V. Ravichandran
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Abstract

We investigate the univalency and the directional convexity of the convolution $\phi\tilde{*}f=\phi*h+\overline{\phi*g}$ of the harmonic mapping $f=h+\bar{g}$ with a mapping $\phi$ whose convolution with the mapping $z+\sum_{k=2}^{\infty}k^nz^k$ is starlike (and such a mapping $\phi$ is called $n$-starlike). In addition, we investigate the directional convexity of (i) the convolution of an analytic convex mapping with the slanted half-plane mapping, and (ii) the partial sums of the convolution of a $6$-starlike mapping with the harmonic Koebe mapping and the harmonic half-plane mapping.

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Subzar Beig, V. Ravichandran. 2017-03-10. Convolution of a harmonic mapping with $n$-starlike mappings and its partial sums. https://arxiv.org/abs/1703.03595

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