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

Let K-nu (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 K-nu (x), and establish several new sharp double inequalities for K nu (x). In particular, the double inequalities & nbsp;(x + a(1))(nu-1/2) < root 2/pi x(nu)e(x) K-nu (x) < (x + b(1))(nu-1/2)& nbsp;(1 + a(2)x)(nu-1/2) < 2(1-nu )/gamma(nu) x(nu)e(x)K(nu) (x) < (1 + b(2)x)(nu-1/2)& nbsp;hold for x > 0 and nu >= 1 with the best constants & nbsp;a(1 & nbsp;)= min{c(0), 1/2 nu + 1/4} and b(1) = max{c(0), 1/2 nu+1/4},& nbsp;& nbsp;a(2) = 1/max {c(0), nu - 1/2} and b(2) = min {c(0), nu - 1/2},& nbsp;where c(0) = 2 (gamma (nu) /root pi )(2/(2 nu-1)).& nbsp;(C) 2021 Elsevier Inc. All rights reserved.