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Zbl 0596.33011
Laforgia, Andrea
Inequalities for Bessel functions. (English)
[J] J. Comput. Appl. Math. 15, 75-81 (1986). ISSN 0377-0427

{\it R. B. Paris}, SIAM J. Math. Anal. 15, 203-205 (1984; Zbl 0536.33006) derived upper and lower bounds for the ratio $J\sb{\nu}(\nu x)/[x\sp{\nu}J\sb{\nu}(\nu)]$ by using recurrence relations and Sonine's integral representation. The present author uses similar techniques to extend such results to general cylinder functions $C\sb{\nu}(x,\alpha)=J\sb{\nu}(x)\cos \alpha -Y\sb{\nu}(x)\sin \alpha$ and to improve the results of Paris in the case $\alpha =0$. Some examples of the author's results run as follows: $$ C\sb{\nu}(\nu t,\alpha) / [C\sb{\nu}(\nu,\alpha)t\sp{\nu}] > \exp (\nu\sp2(1-t\sp2) / (4\nu+4)), \quad \nu>0, \quad 0<t<1, \quad 0\le\alpha\le5\pi/6, $$ and $$ J\sb{\nu}(j'\sb{\nu 1}t) / [J\sb{\nu}(j'\sb{\nu 1})t\sp{\nu}] < exp(j\sp{'2}\sb{\nu 1}(1-t\sp 2)/(2\nu +4)), \quad \nu>0, \quad 0<t<1, $$ where $j'\sb{\nu1}$ is the smallest positive zero of $J'\sb{\nu}(x)$.
[M.E.Muldoon]

MSC 2000:: *33C10 Cylinder functions, etc.
26D99 Inequalities involving real functions

Keywords: Bessel functions; Sonine's integral; cylinder functions

Citations: Zbl 0536.33006

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