Van der Corput inequalities for Bessel functions

Abstract

In this note, we offer some log-concavity properties of certain functions related to Bessel functions of the first kind and modified Bessel functions of the first and second kinds, by solving partially a recent conjecture on the log-convexity/log-concavity properties for modified Bessel functions of the first kind and their derivatives. Moreover, we give an application of the mentioned results by extending two inequalities of van der Corput to Bessel and modified Bessel functions of the first kind. Similar inequalities are proved also for modified Bessel functions of the second kind, as well as for log-concave probability density functions.

Keywords:

• Bessel functions of the first kind
• modified Bessel functions of the first and second kinds
• log-convexity
• van der Corput inequality
• probability density functions
• trigonometricand hyperbolic functions

AMS Subject Classification:

• 33C10
• 26A51
• 26D07
• 39B72

Acknowledgements

The work of Á. Baricz was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. This author is very grateful to Aingeru Fernández from University of the Basque Country, Spain, for pointing out an error in the proof of first theorem in an earlier version of the paper.