Abstract
In the article, we prove that the double inequalities
hold for all \(x>0\) if and only if \(a\geq1/4\) and \(b=0\) if \(a, b\in[0, \infty)\), where \(K_{\nu}(x)\) is the modified Bessel function of the second kind. As applications, we provide bounds for \(K_{n+1}(x)/K_{n}(x)\) with \(n\in\mathbb{N}\) and present the necessary and sufficient condition such that the function \(x\mapsto\sqrt {x+p}e^{x}K_{0}(x)\) is strictly increasing (decreasing) on \((0, \infty)\).
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1 Introduction
The modified Bessel function of the first kind \(I_{\nu}(x)\) is a particular solution of the second-order differential equation
and it can be expressed by the infinite series
While the modified Bessel function of the second kind \(K_{\nu}(x)\) is defined by
where the right-hand side of the identity of (1.1) 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 \(K_{\nu}(x)\) can be found in the literature [1], 9.6.24, 9.6.8, 9.6.9, 9.7.2:
From (1.2) we clearly see that
Recently, the bounds for the modified Bessel function of the second kind \(K_{\nu}(x)\) have attracted the attention of many researchers. Luke [2] proved that the double inequality
holds for all \(x>0\).
Gaunt [3] proved that the double inequality
takes place for all \(x>0\), where \(\Gamma(x)=\int_{0}^{\infty }t^{x-1}e^{-t}\, dx\) is the classical gamma function.
In [4], Segura proved that the double inequality
holds for all \(x>0\) and \(\nu\geq0\).
Bordelon and Ross [5] and Paris [6] provided the inequality
for all \(\nu>-1/2\) and \(y>x>0\).
Laforgia [7] established the inequality
for all \(y>x>0\) if \(\nu\in(0, 1/2)\), and inequality (1.12) is reversed if \(\nu\in(1/2, \infty)\).
Baricz [8] presented the inequality
for all \(y>x>0\) and \(\nu\in(-\infty, -1/2)\cup(1/2, \infty)\).
Motivated by inequality (1.9), in the article, we prove that the double inequality
holds for all \(x>0\) if and only if \(a\geq1/4\) and \(b=0\) if \(a, b\in [0, \infty)\). As applications, we provide bounds for \(K_{n+1}(x)/K_{n}(x)\) with \(n\in\mathbb{N}\) and present the necessary and sufficient condition such that the function \(x\mapsto\sqrt {x+p}e^{x}K_{0}(x)\) is strictly increasing (decreasing) on \((0, \infty)\).
2 Lemmas
In order to prove our main results, we need two lemmas which we present in this section.
Lemma 2.1
See [9]
Let \(-\infty\leq a< b\leq\infty\), \(f,g:[a,b]\rightarrow{\mathbb{R}}\) be continuous on \([a,b]\) and differentiable on \((a,b)\), and \(g'(x)\neq0\) on \((a,b)\). If \(f^{\prime}(x)/g^{\prime}(x)\) is increasing (decreasing) on \((a,b)\), then so are the functions
If \(f^{\prime}(x)/g^{\prime}(x)\) is strictly monotone, then the monotonicity in the conclusion is also strict.
Lemma 2.2
The function
is strictly increasing from \((0, \infty)\) onto \((0, 1/4)\).
Proof
Let \(\omega(t)=\sqrt{(t-1)/(t+1)}\). Then it follows from (1.6), (1.7) and (2.1) that
for all \(x>0\).
Note that (1.3)-(1.5) and (2.1) lead to
3 Main results
Theorem 3.1
Let \(a, b\geq0\). Then the double inequality
holds for all \(x>0\) if and only if \(a\geq1/4\) and \(b=0\).
Proof
Let \(x>0\), \(f(x)\) be defined by Lemma 2.2, and \(f_{1}(x)\), \(f_{2}(x)\) and \(F(x)\) be respectively defined by
and
Then from (1.5), (1.7) and (3.1) we clearly see that
It follows from (3.2)-(3.4), Lemmas 2.1 and 2.2 together with L’Hôpital’s rule that the function \(F(x)\) is strictly increasing on \((0, \infty)\) and
Note that (1.3) and (3.2) lead to
Therefore, Theorem 3.1 follows easily from (3.2), (3.5), (3.6) and the monotonicity of \(F(x)\). □
Remark 3.2
From Lemma 2.2 we clearly see that the double inequality
holds for all \(x>0\) if and only if \(p\leq0\) and \(q\geq1/4\).
From (1.7) and Remark 3.2 we get Corollary 3.3 immediately.
Corollary 3.3
Let \(p, q\geq0\). Then the double inequalities
and
hold for all \(x>0\) if and only if \(p\geq1/4\) and \(q=0\).
Remark 3.4
Let \(p\geq0\). Then from inequality (3.7) we know that the function \(x\mapsto\sqrt{x+p}e^{x}K_{0}(x)\) is strictly increasing (decreasing) on \((0, \infty)\) if and only if \(p=0\) (\(p\geq1/4\)). We clearly see that the bounds for \(K_{1}(x)/K_{0}(x)\) given in Corollary 3.3 are better than the bounds given in (1.10) for \(\nu=0\).
From (1.3), (1.5) and Remark 3.4 we get Corollary 3.5 immediately.
Corollary 3.5
The double inequality
holds for all \(x>0\) if \(p\geq1/4\), and inequality (3.8) is reversed if \(p=0\).
Remark 3.4 also leads to Corollary 3.6.
Corollary 3.6
Let \(p, q\geq0\). Then the double inequality
holds for all \(0< x< y\) if and only if \(p\geq1/4\) and \(q=0\).
Remark 3.7
We clearly see that the lower bound for \(K_{0}(x)/K_{0}(y)\) in Corollary 3.6 is better than the bounds given in (1.11) and (1.12) for \(\nu=0\).
Remark 3.8
From the inequality
given in [10], (2.20), and the fact that
for all \(x>0\) we clearly see that the lower bound given in Theorem 3.1 for \(\sqrt{2/\pi}e^{x}K_{0}(x)\) is better than that given in (1.8) and (1.9). But the upper bound given in Theorem 3.1 is weaker than that given in (1.8).
Remark 3.9
From Theorem 3.1 and Corollary 3.3 we clearly see that there exist \(\theta_{1}=\theta_{1}(x)\in(0, 1/4)\) and \(\theta_{2}=\theta_{2}(x)\in (0, 1/4)\) such that
for all \(x>0\).
Theorem 3.10
Let \(x>0\), \(n\in\mathbb{N}\), \(R_{n}(x)=K_{n+1}(x)/K_{n}(x)\), \(u_{0}(x)=1+1/(2x)\), \(v_{0}(x)=1+1/(2x+1/2)\), and \(u_{n}(x)\) and \(v_{n}(x)\) be defined by
Then the double inequality
holds for all \(x>0\) and \(n\in\mathbb{N}\).
Proof
We use mathematical induction to prove inequality (3.10). From Corollary 3.3 we clearly see that inequality (3.10) holds for all \(x>0\) and \(n=0\).
Suppose that inequality (3.10) holds for \(n=k-1\) (\(k\geq1\)), that is,
Then it follows from (3.9) and (3.11) together with the formula
given in [11] that
Inequality (3.12) implies that inequality (3.10) holds for \(n=k\), and the proof of Theorem 3.10 is completed. □
Remark 3.11
Let \(n=1, 2, 3\). Then Theorem 3.10 leads to
for all \(x>0\).
Remark 3.12
It is not difficult to verify that
for \(x>0\). Therefore, the bounds given in Remark 3.11 are better than the bounds given in (1.10) for \(\nu=1, 2, 3\).
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Acknowledgements
The research was supported by the Natural Science Foundation of China under Grants 61673169, 61374086, 11371125 and 11401191.
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Yang, ZH., Chu, YM. On approximating the modified Bessel function of the second kind. J Inequal Appl 2017, 41 (2017). https://doi.org/10.1186/s13660-017-1317-z
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DOI: https://doi.org/10.1186/s13660-017-1317-z