International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 3, Pages 561-566

A note on monotonicity property of Bessel functions

Stamatis Koumandos1,2

1Department of Pure Mathematics, The University of Adelaide, Adelaide 5005, Australia
2Department of Mathematics and Statistics, University of Cyprus, P.O. Box 537, Nicosia 1678, Cyprus

Received 18 September 1995

Copyright © 1997 Stamatis Koumandos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A theorem of Lorch, Muldoon and Szegö states that the sequence {jα,kjα,k+1tα|Jα(t)|dt}k=1 is decreasing for α>1/2, where Jα(t) the Bessel function of the first kind order α and jα,k its kth positive root. This monotonicity property implies Szegö's inequality 0xtαJα(t)dt0, when αα and α is the unique solution of 0jα,2tαJα(t)dt=0.

We give a new and simpler proof of these classical results by expressing the above Bessel function integral as an integral involving elementary functions.