On approximating the modified Bessel function of the second kind

Abstract

Abstract In the article, we prove that the double inequalities π e − x 2 ( 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 K ν ( x ) $K_{\nu}(x)$ is the modified Bessel function of the second kind. As applications, we provide bounds for K n + 1 ( x ) / K n ( 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 + p e x K 0 ( x ) $x\mapsto\sqrt {x+p}e^{x}K_{0}(x)$ is strictly increasing (decreasing) on ( 0 , ∞ ) $(0, \infty)$ .

Keywords

• modified Bessel function
• gamma function
• monotonicity