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Close-to-Convexity of Some Special Functions and Their Derivatives

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Abstract

In this paper our aim is to deduce some sufficient (and necessary) conditions for the close-to-convexity of some special functions and their derivatives, like Bessel functions, Struve functions, and a particular case of Lommel functions of the first kind, which can be expressed in terms of the hypergeometric function \({}_1F_2\). The key tool in our proofs is a result of Shah and Trimble about transcendental entire functions with univalent derivatives. Moreover, a known result of Pólya on entire functions, the infinite product representations and some results on zeros of Bessel, Struve, and Lommel functions of the first kind are used in order to achieve the main results of the paper.

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Acknowledgments

The research of Á. Baricz was supported by a research grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project Number PN-II-RU-TE-2012-3-0190/2014.

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Authors and Affiliations

  1. Department of Economics, Babeş-Bolyai University, 400591, Cluj-Napoca, Romania

    Árpád Baricz

  2. Department of Mathematics and Informatics, Sapientia Hungarian University of Transylvania, 540485, Târgu-Mureş, Romania

    Róbert Szász

Authors
  1. Árpád Baricz
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  2. Róbert Szász
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Correspondence to Árpád Baricz.

Additional information

Communicated by V. Ravichandran.

Á. Baricz dedicates this paper to his friend Prof. Tibor K. Pogány on the occasion of his 60th birthday.

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Baricz, Á., Szász, R. Close-to-Convexity of Some Special Functions and Their Derivatives. Bull. Malays. Math. Sci. Soc. 39, 427–437 (2016). https://doi.org/10.1007/s40840-015-0180-7

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  • DOI: https://doi.org/10.1007/s40840-015-0180-7

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