LAUSR

In the article, we prove that the double inequalities πe−x2(x+a)0$x>0$ if and only if a≥1/4$a\geq1/4$ and b=0$b=0$ if a,b∈[0,∞)$a, b\in[0, \infty)$, where Kν(x)$K_{\nu}(x)$ is the modified Bessel function of the second kind. As applications, we provide bounds for Kn+1(x)/Kn(x)$K_{n+1}(x)/K_{n}(x)$ with n∈N$n\in\mathbb{N}$ and present the necessary and sufficient condition such that the function x↦x+pexK0(x)$x\mapsto\sqrt {x+p}e^{x}K_{0}(x)$ is strictly increasing (decreasing) on (0,∞)$(0, \infty)$.