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On De la Vallée Poussin means for harmonic mappings

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Abstract

In this article, we study the geometric properties of \(V_n(f)\), the \(n^{th}\) De la Vallée Poussin means for univalent starlike harmonic mappings f. In particular, we provide a necessary and sufficient condition for \(V_n(f)\) to be univalent and starlike in the unit disk \({\mathbb {D}}\), when \(f \in {\mathcal S}_H^{*}\), the class of all normalized univalent starlike harmonic mappings in \({\mathbb {D}}\). We determine the radius of fully starlikeness (respectively, fully convexity) of \(V_2(f)\), when \(f \in {{\mathcal {S}}}_H^{0}\) and the result is sharp. Then, we determine the radius \(r_n \in (0, 1)\) so that \(V_n(f)\) is univalent and fully starlike in \(|z| < r_n\), whenever f is univalent and fully starlike harmonic mapping in \({\mathbb {D}}\). We also discuss about the geometry preserving nature of \(V_n(f)\), when f belongs to some well known geometric subclasses of \({{\mathcal {S}}}_H\).

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Acknowledgements

I dedicate this research article to my parents Mr. A. Anbareeswaran and Mrs. A. Neelapushpam. This research work was supported by the ISIRD Project (F.No. 9-308/2018/IITRPR/4092) of Indian Institute of Technology Ropar, India.

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  1. Indian Institute of Technology Ropar, Nangal Road, Rupnagar, Punjab, 140001, India

    Anbareeswaran Sairam Kaliraj

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Communicated by Adrian Constantin.

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Kaliraj, A.S. On De la Vallée Poussin means for harmonic mappings. Monatsh Math 198, 547–564 (2022). https://doi.org/10.1007/s00605-021-01660-3

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