On approximating the modified Bessel function of the second kind

In the article, we prove that the double inequalities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\sqrt{\pi}e^{-x}}{\sqrt{2(x+a)}}< K_{0}(x)< \frac{\sqrt{\pi }e^{-x}}{\sqrt{2(x+b)}},\qquad 1+ \frac{1}{2(x+a)}< \frac {K_{1}(x)}{K_{0}(x)}< 1+\frac{1}{2(x+b)} $$\end{document}πe−x2(x+a)0$\end{document}x>0 if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a\geq1/4$\end{document}a≥1/4 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b=0$\end{document}b=0 if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a, b\in[0, \infty)$\end{document}a,b∈[0,∞), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K_{\nu}(x)$\end{document}Kν(x) is the modified Bessel function of the second kind. As applications, we provide bounds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K_{n+1}(x)/K_{n}(x)$\end{document}Kn+1(x)/Kn(x) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\in\mathbb{N}$\end{document}n∈N and present the necessary and sufficient condition such that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\mapsto\sqrt {x+p}e^{x}K_{0}(x)$\end{document}x↦x+pexK0(x) is strictly increasing (decreasing) on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(0, \infty)$\end{document}(0,∞).


Introduction
The modified Bessel function of the first kind I ν (x) is a particular solution of the secondorder differential equation x  y (x) + xy (x)x  + ν  y(x) = , and it can be expressed by the infinite series While the modified Bessel function of the second kind K ν (x) is defined by where the right-hand side of the identity of (.) is the limiting value in case ν is an integer.
The following integral representation formula and asymptotic formulas for the modified Bessel function of the second kind K ν (x) can be found in the literature [], .., .., .., ..: From (.) we clearly see that Recently, the bounds for the modified Bessel function of the second kind K ν (x) have attracted the attention of many researchers. Luke [] proved that the double inequality holds for all x > . Gaunt [] proved that the double inequality takes place for all x > , where (x) = ∞  t x- e -t dx is the classical gamma function. In [], Segura proved that the double inequality holds for all x >  and ν ≥ .
Motivated by inequality (.), in the article, we prove that the double inequality . As applications, we provide bounds for K n+ (x)/K n (x) with n ∈ N and present the necessary and sufficient condition such that the function x → √ x + pe x K  (x) is strictly increasing (decreasing) on (, ∞).

Lemmas
In order to prove our main results, we need two lemmas which we present in this section. .
is strictly monotone, then the monotonicity in the conclusion is also strict.

Main results
Theorem . Let a, b ≥ . Then the double inequality holds for all x >  if and only if a ≥ / and b = .
Proof Let x > , f (x) be defined by Lemma ., and f  (x), f  (x) and F(x) be respectively defined by and Then from (.), (.) and (.) we clearly see that Remark . From Lemma . we clearly see that the double inequality and We clearly see that the bounds for K  (x)/K  (x) given in Corollary . are better than the bounds given in (.) for ν = .

Corollary . The double inequality
holds for all x >  if p ≥ /, and inequality (.) is reversed if p = .
Remark . also leads to Corollary ..
Corollary . Let p, q ≥ . Then the double inequality holds for all  < x < y if and only if p ≥ / and q = .

Remark .
We clearly see that the lower bound for K  (x)/K  (y) in Corollary . is better than the bounds given in (.) and (.) for ν = .

Remark . From the inequality
given in [], (.), and the fact that for all x >  we clearly see that the lower bound given in Theorem . for √ /πe x K  (x) is better than that given in (.) and (.). But the upper bound given in Theorem . is weaker than that given in (.).
Proof We use mathematical induction to prove inequality (.). From Corollary . we clearly see that inequality (.) holds for all x >  and n = . Suppose that inequality (.) holds for n = k - (k ≥ ), that is, Then it follows from (.) and (.) together with the formula Inequality (.) implies that inequality (.) holds for n = k, and the proof of Theorem . is completed.
Remark . Let n = , , . Then Theorem . leads to x for x > . Therefore, the bounds given in Remark . are better than the bounds given in (.) for ν = , , .