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On the Square of the First Zero of the Bessel Function Jv(z)

Published online by Cambridge University Press:  20 November 2018

Árpád Elbert  and
Panayiotis D. Siafarikas
Árpád Elbert
Affiliation:
Mathematical Institute of the Hungarian Academy of Science Budapest POB. 428 1376 Hungary
Panayiotis D. Siafarikas
Affiliation:
Department of Mathematics University of Patras Patras Greece
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Abstract

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Let ${{j}_{v,1}}$ be the smallest (first) positive zero of the Bessel function ${{J}_{v}}(z),\,v\,>\,-\,1$, which becomes zero when $v$ approaches −1. Then $j_{v,1}^{2}$ can be continued analytically to $-2\,<\,v\,<\,-1$, where it takes on negative values. We show that $j_{v,1}^{2}$ is a convex function of $v$ in the interval $-2\,<\,v\,\le \,0$, as an addition to an old result [Á. Elbert and A. Laforgia, SIAM J. Math. Anal. 15(1984), 206–212], stating this convexity for $v\,>\,0$. Also the monotonicity properties of the functions $\frac{j_{v,1}^{2}}{4(v+1)},\,\frac{j_{v,1}^{2}}{4(v+1)\sqrt{v+2}}$ are determined. Our approach is based on the series expansion of Bessel function ${{J}_{v}}(z)$ and it turned out to be effective, especially when $-2\,<\,v\,<\,-1$.

Keywords

33A40
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 1 , 01 March 1999 , pp. 56 - 67
Copyright
Copyright © Canadian Mathematical Society 1999

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