On the Square of the First Zero of the Bessel Function $J_\nu(z)$

http://dx.doi.org/10.4153/CMB-1999-007-4
Canad. Math. Bull. 42(1999), 56-67

  Published:1999-03-01
 Printed: Mar 1999

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