Monotonicity of the ratio of modified Bessel functions of the first kind with applications
- Zhen-Hang Yang^{1, 2} and
- Shen-Zhou Zheng^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-018-1648-4
© The Author(s) 2018
Received: 6 December 2017
Accepted: 20 February 2018
Published: 9 March 2018
Abstract
Keywords
MSC
1 Introduction
Theorem 1.1
- (i)
If \(p\geq v+1/2\) for \(v\geq-3/2\) or \(p\geq c_{v}\) for \(-2< v<-3/2\), then the function \(F_{p}\) is increasing from \(( 0,\infty ) \) onto \(( ( 2v+2-p ) / ( v+2 ) ,1 ) \).
- (ii)
If \(p\leq v-1\), then \(F_{p}\) is decreasing from \(( 0,\infty ) \) onto \(( 1, ( 2v+2-p ) / ( v+2 ) ) \).
- (iii)If \(v-1< p< v+1/2\), then there exists an \(x_{0}>0\) such that \(F_{p}\) is increasing on \(( 0,x_{0} ) \), and decreasing on \(( x_{0},\infty ) \). Consequently, it holds that for \(x>0\),where \(\lambda_{p}=F_{p} ( x_{0} ) \), and \(x_{0}\) is a unique solution of the equation \(F_{p}^{\prime} ( x ) =0\) on \(( 0,\infty ) \).$$ \min \biggl\{ \frac{2v+2-p}{v+2},1 \biggr\} < F_{p} ( x ) < \lambda_{p}, $$(1.8)
- (iv)If \(v+1/2\leq p< c_{v}\) for \(-2< v<-3/2\), then we havefor \(x>0\), where \(\theta_{p}=\sup_{x>0}F_{p} ( x ) \). The lower and upper bounds for \(F_{p} ( x ) \) are sharp.$$ \frac{2v+2-p}{v+2}< F_{p} ( x ) < \theta_{p} $$(1.9)
The rest of this paper is organized as follows. In Sect. 2, some lemmas are listed. The proof of Theorem 1.1 is presented in Sect. 3. In Sect. 4, as applications of Theorem 1.1, some Simpson–Spector-type inequalities for \(W_{v} ( x ) \) are established in Sect. 4.1; in Sect. 4.2, a new type of bounds \(p+r\sqrt{x^{2}+q^{2}}\) (\(r>0\)) for \(W_{v} ( x ) \) for \(p<2v+2\) with \(v>-2\) is established, and a new Amos-type upper bound \(p+\sqrt{x^{2}+q^{2}}\) for \(-2< v<-3/2\) is presented; some computable bounds for \(W_{v} ( x ) \) for \(v-1< p<\min \{ v+1/2,2v+2 \} \) with \(v>-2\) and for \(2v+2< p< v+1/ ( 2v+5 ) \) with \(-2< v<-3/2\) are found in Sect. 4.3.
2 Lemmas
To prove Theorem 1.1, we need some lemmas. The following lemma which comes from [19, (3.5)] (see also [20]) is useful.
Lemma 2.1
Lemma 2.2
([21])
Let \(A ( x ) =\sum_{k=0}^{\infty}a_{k}x^{k}\) and \(B ( x ) =\sum_{k=0}^{\infty }b_{k}x^{k}\) be two real power series converging on \(( -r,r ) \) for some \(r>0\) with \(b_{k}>0\) for all k. If the sequence \(\{ a_{k}/b_{k}\}\) is increasing (or decreasing) for all k, then the function \(x\mapsto A ( x ) /B ( x ) \) is also increasing (or decreasing) on \(( 0,r ) \).
Lemma 2.2 is a powerful tool to deal with the monotonicity of the ratio between two power series. An improvement of Lemma 2.2 has been presented in [22, Theorem 2.1]. A similar monotonicity rule for the ratio of two Laplace transforms was established in [23, Lemma 4] (see also [24]).
Lemma 2.3
Let \(A ( x ) =\sum_{k=0}^{\infty }a_{k}x^{k}\) and \(B ( x ) =\sum_{k=0}^{\infty}b_{k}x^{k}\) be two real power series converging on \(\mathbb{R}\) with \(b_{k}>0\) for all k. If, for certain \(m\in \mathbb{N}\), the non-constant sequence \(\{a_{k}/b_{k}\}\) is increasing (or decreasing) for \(0\leq k\leq m\) and decreasing (or increasing) for \(k\geq m\), then there is a unique \(x_{0}\in ( 0,\infty ) \) such that the function \(A/B\) is increasing (or decreasing) on \(( 0,x_{0} ) \) and decreasing (or increasing) on \(( x_{0},\infty ) \).
Lemma 2.3 first appeared in [25, Lemma 6.4] without giving the details of the proof. Two strict proofs were given in [22] and [26]. Another useful tool associated with Lemma 2.3 is the sign rule of a class of special series or polynomials, see, for example, [25, Lemma 6.3], [27, Lemma 7], [28]).
Lemma 2.4
([29, Problems 85, 94])
3 Proof of Theorem 1.1
Now we are in a position to prove Theorem 1.1.
Proof
Now we discuss the monotonicity of \(F_{p}\) by dividing it into two cases.
Case 1. \(v\geq-3/2\).
Subcase 1.1. \(p\geq g_{\infty} ( v ) =v+1/2\). From relation (3.3) it is obtained that the sequence \(\{ a_{n}/b_{n}\}_{n\geq1}\) is increasing. By Lemma 2.2 it follows that the ratio \(f_{1}/f_{2}\) is increasing on \(( 0,\infty ) \).
Subcase 1.2. \(p\leq g_{1} ( v ) =v-1\). It is seen that the sequence \(\{a_{n}/b_{n}\}_{n\geq1}\) is decreasing, and from Lemma 2.2 it follows that the ratio \(f_{1}/f_{2}\) is decreasing on \(( 0,\infty ) \).
Case 2. \(-2< v<-3/2\).
Subcase 2.1. \(p\geq g_{2} ( v ) =c_{v}\). In the same way, we get that the ratio \(f_{1}/f_{2}\) is increasing on \(( 0,\infty ) \).
Subcase 2.2. \(p\leq g_{1} ( v ) =v-1\). Similarly, we find that the ratio \(f_{1}/f_{2}\) is decreasing on \(( 0,\infty ) \).
The continuity of the function \(F_{p} ( x ) \) on \(( 0,\infty ) \) together with \(F_{p} ( 0 ) =a_{1}/b_{1}\) and \(F_{p} ( \infty ) =1\) means that \(F_{p} ( x ) \) is bounded on \(( 0,\infty ) \), so \(\sup_{x>0}F_{p} ( x ) \) exists, which completes the proof. □
Remark 3.1
Clearly, we are not able to describe the monotone pattern of \(f_{1}/f_{2}\) by directly using Lemmas 2.2 and 2.3. We here guess that there are two \(x_{1}\), \(x_{2}\) with \(x_{2}>x_{1}>0\) such that \(f_{1}/f_{2}\) is increasing on \(( 0,x_{1} ) \cup ( x_{2},\infty ) \) and decreasing on \(( x_{1},x_{2} ) \).
4 Some new type of bounds for \(W_{v} ( x ) \)
4.1 Simpson–Spector-type inequality for \(W_{v} ( x ) \)
Proposition 4.1
Proof
(ii) The necessary and sufficient conditions for the right-hand side inequality of (4.1) to hold are obvious.
This completes the proof. □
Remark 4.2
In addition, putting \(p=c_{v}\) with \(-2< v<-3/2\) in Proposition 4.1, where \(c_{v}\) is given in (1.7), we obtain a new Simpson–Spector-type inequality, which is stated as a corollary.
Corollary 4.3
4.2 Sharp bounds for \(W_{v} ( x ) \) in the form of \(p+r\sqrt{x^{2}+q^{2}}\)
Proposition 4.4
Proof
(i) By Proposition 4.1, the necessary and sufficient condition such that the left-hand side inequality of (4.5) holds for \(x>0\) and \(( p,v ) \in E_{0}\) is clear.
This completes the proof. □
Remark 4.5
Amos [10, Eq. (11)] offered a lower bound \(A_{v,v+2} ( x ) < W_{v} ( x ) \) for \(x>0\) and \(v\geq0\). Hornik and Grün [9, Theorem 6] showed that the Amos-type bound is the sharpest for \(x>0\) and \(v>-1\). Yang and Zheng [6, Theorem 4.6] extended the range of v from \(v>-1\) to \(v>-3/2\). Proposition 4.4 presents another lower bound \(L_{p} ( x ) \) defined in (4.6) for \(W_{v} ( x ) \) for \(x>0\) with \(v>-2\) and shows that \(L_{v} ( x ) \) defined by (4.8) is the maximum over all lower bounds \(\{ L_{p} ( x ) :p<2v+2,v>-2 \} \). It should be emphasized that our sharpest lower bound \(L_{v} ( x ) \) extends the range of \(A_{v,v+2} ( x ) \) from \(v>-3/2\) to \(v>-2\) although \(L_{v} ( x ) \) and \(A_{v,v+2} ( x ) \) have the same expression.
Remark 4.6
Amos [10, Eq. (16)] gave an upper bound \(W_{v} ( x ) < A_{v+1/2,v+3/2} ( x ) \) for \(x>0\) and \(v\geq0 \). Hornik and Grün [9, Theorem 3] proved that this Amos-type upper bound is the best for \(x>0\) and \(v>-1\), where the range of v has been extended from \(v>-1\) to \(v>-3/2\) in [6, Theorem 4.4] by Yang and Zheng. Since \(U_{v+1/2}^{ ( 1 ) } ( x ) =A_{v+1/2,v+3/2} ( x ) \), our Proposition 4.4 demonstrates the same result in [6, Theorem 4.4] by a slightly different approach.
Remark 4.7
As a direct consequence of Proposition 4.4, we have the following.
Corollary 4.8
Remark 4.9
The right-hand side in inequality (4.15) for \(v\geq 0\) was proved in [10, Eq. (16)] by Amos, and for \(v>-3/2\) it follows from Neuman’s inequality (1.4). The right-hand side one in (4.16) for \(v\geq0\) is also due to Amos [10, Eq. (11)], which for \(v>-1\) was proved by Yuan and Kalbfleisch [12, Eq. (A.5)], and Laforgia and Natalini [14, Theorem 1.1]. While the left-hand side inequality in (4.16) for \(v>-1\) was showed by Segura [15, Eq. (61)]. Inequalities (4.17) were proved by Yang and Zheng in [6, Remark 4.9]. Moreover, the rational bounds given in (4.18) appeared in [4, Appendix] for \(v>-1\), so the right-hand side inequality of (4.18) can be viewed as a new one in the sense that the range of v is extended from \(v>-1\) to \(v>-2\).
Proposition 4.10
Let \(-2< v<-3/2\). Then the double inequality (4.19) holds for \(x>0\) and \(p\geq c_{v}^{\ast}=v+1/ ( 2v+5 ) \) with the best constant \(\beta=1\).
Proof
(i) For \(p\geq c_{v}\), the desired result is evidently valid by Proposition 4.1.
Remark 4.11
It is easy to check that the lower bound for \(W_{v} ( x ) \) given in (4.19) is weaker than \(2v+2\), but the upper bound for \(\beta=1\) is clearly a new Amos-type bound for \(p\geq c_{v}^{\ast}\) with \(-2< v<-3/2\) since it is not comparable with the sharpest upper bound \(U_{v-1}^{ ( 2 ) } ( x ) \) for \(p\geq c_{v}^{\ast}\) with \(-2< v<-3/2\), while another one \(U_{v+1/2}^{ ( 1 ) } ( x ) \) is restricted in \(v\geq-3/2\).
4.3 Some computable bounds for \(W_{v} ( x ) \)
Proposition 4.12
Proof
Now by Proposition 4.12 we have the following.
Corollary 4.13
- (i)For \(v-1< p\leq v-1/2\), the inequalityholds for \(x>0\).$$ W_{v} ( x ) < p+\sqrt{\frac{v+3}{v+2}x^{2}+ ( 2v+2-p ) ^{2}}=U_{p}^{\ast\ast} ( x ) $$(4.22)
- (ii)For \(v-1/2< p<\min \{ v+1/2,2v+2 \} \), the inequalityholds for \(x>0\). In particular, taking \(p=v\) and letting \(p\rightarrow v+1/2 \) with \(v\geq-3/2\) and \(p\rightarrow2v+2\) with \(-2< v<-3/2\), the following inequalities hold for \(x>0\):$$ W_{v} ( x ) < p+\sqrt{\frac{4v+5-2p}{2 ( v+2 ) }x^{2}+ ( 2v+2-p ) ^{2}}=U_{p}^{\ast} ( x ) $$(4.23)$$\begin{aligned}& W_{v} ( x ) < v+\sqrt{\frac{2v+5}{2 ( v+2 ) }x^{2}+ ( v+2 ) ^{2}}=U_{v}^{\ast} ( x )\quad \textit{for }v>-2, \end{aligned}$$(4.24)$$\begin{aligned}& W_{v} ( x ) < v+\frac{1}{2}+\sqrt{x^{2}+ \biggl( v+ \frac {3}{2} \biggr) ^{2}}=U_{v+1/2}^{\ast} ( x ) \quad \textit{for }v\geq-\frac{3}{2}, \end{aligned}$$(4.25)$$\begin{aligned}& W_{v} ( x ) < 2v+2+\frac{x}{\sqrt{2v+4}}=U_{2v+2}^{\ast } (x )\quad \textit{for }-2< v< -\frac{3}{2}. \end{aligned}$$(4.26)
Remark 4.14
Remark 4.15
It thus can be seen that the upper bound \(U_{p}^{\ast} ( x ) \) for \(v-1/2< p<\min \{ v+1/2,2v+2 \} \) with \(v>-2\) belongs to the new type of bounds \(p+r\sqrt{x^{2}+q^{2}}\) (\(r>0\)) for \(W_{v} ( x ) \).
Proposition 4.16
Proposition 4.17
5 Conclusions
As more important applications, we reproved some known results and also found a new type of bounds \(p+r\sqrt{x^{2}+q^{2}}\) for \(W_{v} ( x ) \).
(v) Using the same technique as Proposition 4.12, we established two new double inequalities for \(W_{v} ( x ) \) in the cases of \(2v+2< p< v+1/2\) and \(v+1/2\leq p< c_{v}^{\ast}\) for \(-2< v<-3/2\), that are, (4.27) and (4.29). However, the lower bounds given in (4.27) and (4.29) for \(W_{v} ( x ) \) are trivial since they are weaker than \(2v+2\). The upper bounds belong to the type of bounds \(p+r\sqrt{x^{2}+q^{2}}\) (\(r>0\)).
Additionally, as a consequence of our results, we deduced some new inequalities for \(W_{v} ( x ) \), for example, (4.18), (4.26), and also reobtained some known important inequalities, such as the inequalities proved by Amos [10], Yuan and Kalbfleisch [12, (A.5)], Laforgia and Natalini [14, Theorem 1.1], Segura [15, (61)], [4, Appendix] and so on.
Declarations
Acknowledgements
This paper is supported by the National Science Foundation of China grant No. 11371050.
Authors’ contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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