Monotonicity and inequalities involving the modified Bessel functions of the second kind

Abstract

Let Kν(x) be the modified Bessel functions of the second kind. In this paper, we give the monotonicity and complete monotonicity results for several functions involving Kν(x), and establish several new sharp double inequalities for Kν(x). In particular, the double inequalities(x+a1)ν−1/2<2πxνexKν(x)<(x+b1)ν−1/2(1+a2x)ν−1/2<21−νΓ(ν)xνexKν(x)<(1+b2x)ν−1/2hold for x>0 and ν≥1 with the best constantsa1=min⁡{c0,12ν+14} and b1=max⁡{c0,12ν+14},a2=1max⁡{c0,ν−1/2} and b2=1min⁡{c0,ν−1/2},where c0=2(Γ(ν)/π)2/(2ν−1).