A note on monotonicity property of Bessel functions

Source abstract

A theorem of Lorch, Muldoon and Szegö states that the sequence urn:x-wiley:01611712:media:ijmm683727:ijmm683727-math-0001 is decreasing for α > −1/2, where J α ( t ) the Bessel function of the first kind order α and j α , k its k th positive root. This monotonicity property implies Szegö′s inequality urn:x-wiley:01611712:media:ijmm683727:ijmm683727-math-0002 , when α ≥ α ′ and α ′ is the unique solution of . We give a new and simpler proof of these classical results by expressing the above Bessel function integral as an integral involving elementary functions.