# Radius Problems for Functions Containing Derivatives of Bessel Functions

Radius Problems for Functions Containing Derivatives of Bessel Functions In this paper our aim is to find the radii of starlikeness and convexity for three different kinds of normalizations of the function Nν(z)=az2Jν″(z)+bzJν′(z)+cJν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_\nu (z)=az^{2}J_{\nu }^{\prime \prime }(z)+bzJ_{\nu }^{\prime }(z)+cJ_{\nu }(z)$$\end{document}, where Jν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$J_\nu (z)$$\end{document} is the Bessel function of the first kind of order ν\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nu$$\end{document}. The key tools in the proof of our main results are the Mittag-Leffler expansion for the function Nν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_\nu (z)$$\end{document} and properties of real zeros of it. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized function Nν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_\nu (z)$$\end{document}. Finally, we evaluate certain multiple sums of the zeros for the function Nν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_\nu (z)$$\end{document}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# Radius Problems for Functions Containing Derivatives of Bessel Functions

, Volume OnlineFirst – May 31, 2022
26 pages

Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-022-00455-3
Publisher site
See Article on Publisher Site

### Abstract

In this paper our aim is to find the radii of starlikeness and convexity for three different kinds of normalizations of the function Nν(z)=az2Jν″(z)+bzJν′(z)+cJν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_\nu (z)=az^{2}J_{\nu }^{\prime \prime }(z)+bzJ_{\nu }^{\prime }(z)+cJ_{\nu }(z)$$\end{document}, where Jν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$J_\nu (z)$$\end{document} is the Bessel function of the first kind of order ν\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nu$$\end{document}. The key tools in the proof of our main results are the Mittag-Leffler expansion for the function Nν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_\nu (z)$$\end{document} and properties of real zeros of it. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized function Nν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_\nu (z)$$\end{document}. Finally, we evaluate certain multiple sums of the zeros for the function Nν(z)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N_\nu (z)$$\end{document}.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: May 31, 2022

Keywords: Normalized Bessel functions of the fist kind; Convex functions; Starlike functions; Zeros of Bessel function derivatives; Radius; Primary 33C10; Secondary 30C45

### References

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