Geodesic
Fully starlike and fully convex harmonic mappings of order $\alpha$
Sumit Nagpal1 ; V. Ravichandran2
1 Department of Mathematics University of Delhi Delhi 110 007, India
2 Department of Mathematics University of Delhi Delhi 110 007, India and School of Mathematical Sciences Universiti Sains Malaysia 11800 USM, Penang, Malaysia
Annales Polonici Mathematici, Tome 108 (2013) no. 1, pp. 85-107

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

The hereditary properties of convexity and starlikeness for conformal mappings do not generalize to univalent harmonic mappings. This failure leads to the notions of fully starlike and fully convex mappings. In this paper, properties of fully starlike mappings of order $\alpha$ and fully convex mappings of order $\alpha$ $(0\leq \alpha 1)$ are studied; in particular, the bounds for the radius of full starlikeness of order $\alpha$ as well as the radius of full convexity of order $\alpha$ are determined for certain families of univalent harmonic mappings. Unlike the analytic case, convexity is not preserved under the convolution of univalent harmonic convex mappings. Given two univalent harmonic convex mappings $f$ and $g$, the radius $r_{0}$ such that their harmonic convolution $f*g$ is a univalent harmonic convex mapping in $|z| r_{0}$ is also investigated.
DOI : 10.4064/ap108-1-7
Keywords: hereditary properties convexity starlikeness conformal mappings generalize univalent harmonic mappings failure leads notions fully starlike fully convex mappings paper properties fully starlike mappings order alpha fully convex mappings order alpha leq alpha studied particular bounds radius full starlikeness order alpha radius full convexity order alpha determined certain families univalent harmonic mappings unlike analytic convexity preserved under convolution univalent harmonic convex mappings given univalent harmonic convex mappings radius their harmonic convolution f*g univalent harmonic convex mapping investigated

Sumit Nagpal 1 ; V. Ravichandran 2

1 Department of Mathematics University of Delhi Delhi 110 007, India
2 Department of Mathematics University of Delhi Delhi 110 007, India and School of Mathematical Sciences Universiti Sains Malaysia 11800 USM, Penang, Malaysia 
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Sumit Nagpal; V. Ravichandran. Fully starlike and fully convex harmonic mappings of order  $\alpha$. Annales Polonici Mathematici, Tome 108 (2013) no. 1, pp. 85-107. doi: 10.4064/ap108-1-7

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