Geodesic
On the Square of the First Zero of the Bessel Function Jv(z)
Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 56-67

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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 $-2\,<\,v\,<\,-1$ .
DOI : 10.4153/CMB-1999-007-4
Mots-clés : 33A40 
Elbert, Árpád; Siafarikas, Panayiotis D. On the Square of the First Zero of the Bessel Function Jv(z). Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 56-67. doi: 10.4153/CMB-1999-007-4
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     title = {On the {Square} of the {First} {Zero} of the {Bessel} {Function} {Jv(z)}},
     journal = {Canadian mathematical bulletin},
     pages = {56--67},
     year = {1999},
     volume = {42},
     number = {1},
     doi = {10.4153/CMB-1999-007-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-007-4/}
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