International Journal of Mathematics and Mathematical Sciences 
Volume 20 (1997), Issue 3, Pages 561-566
doi:10.1155/S0161171297000756

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

Abstract

A theorem of Lorch, Muldoon and Szegö states that the sequence {∫jα,kjα,k+1t−α|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 ∫0xt−αJα(t)dt≥0, when α≥α′ and α′ is the unique solution of ∫0jα,2t−αJα(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.