Bounds for Tur\'anians of modified Bessel functions

TL;DR

This paper derives new tight bounds for Turán-type inequalities involving modified Bessel functions, with applications demonstrating sharp inequalities and properties like geometric concavity, relevant across various scientific fields.

Contribution

It introduces novel tight Turán inequalities for modified Bessel functions, improving bounds and revealing geometric concavity of their product, using advanced differential equation and integral techniques.

Findings

Sharp bounds for Turánian of modified Bessel functions

The product of modified Bessel functions is strictly geometrically concave

Bounds' relative errors tend to zero as argument increases

Abstract

Motivated by some applications in applied mathematics, biology, chemistry, physics and engineering sciences, new tight Tur\'an type inequalities for modified Bessel functions of the first and second kind are deduced. These inequalities provide sharp lower and upper bounds for the Tur\'anian of modified Bessel functions of the first and second kind, and in most cases the relative errors of the bounds tend to zero as the argument tends to infinity. The chief tools in our proofs are some ideas of Gronwall [19] on ordinary differential equations, an integral representation of Ismail [28,29] for the quotient of modified Bessel functions of the second kind and some results of Hartman and Watson [24,26,59]. As applications of the main results some sharp Tur\'an type inequalities are presented for the product of modified Bessel functions of the first and second kind and it is shown that this product is strictly geometrically concave.