On approximating the modified Bessel function of the second kind

Abstract

In the article, we prove that the double inequalities πe−x2(x+a) <1+12(x+b) hold for all x> 0 if and only if a≥ 1/4 and b= 0 if a, b∈ [0 , ∞) , where Kν(x) is the modified Bessel function of the second kind. As applications, we provide bounds for Kn+1(x)/Kn(x) with n∈ N and present the necessary and sufficient condition such that the function x↦x+pexK0(x) is strictly increasing (decreasing) on (0 , ∞).