Inequalities for the modified Bessel function of the second kind and the kernel of the Krätzel integral transformation

Abstract

Abstract. We obtain new inequalities for the modified Bessel function of the second kind K ν in terms of the gamma function. These bounds follow as special cases of inequalities that we derive for the kernel of the Krätzel integral transformation.Mathematics subject classification (2010): 33C10.

Certain inequalities for the modified Bessel-type function

“…where Q β,β/p,νp (x) and Γ (α, x) are respectively defined by (4) and (8). The equality in (44) holds if and only if ν = 1 β and σ = 0.…”

“…If ν > 1 n , the strict inequality is reversed and holds for all x > (n -1)(ν - 1 n ). By setting n = 2 in (2), Gaunt [8] obtained a lower bound for the modified Bessel function K ν (x), that is,…”