Inequalities on an extended Bessel function
- Rosihan M. Ali1,
- See Keong Lee1Email author and
- Saiful R. Mondal2
https://doi.org/10.1186/s13660-018-1656-4
© The Author(s) 2018
Received: 25 November 2017
Accepted: 10 March 2018
Published: 27 March 2018
Abstract
Keywords
MSC
1 Introduction
Apparently not much has been investigated for the extended Bessel function given by (1). Presumably such extensions would readily follow from recent results along similar used arguments, albeit involving intense computations. Still several pertinent questions remain, which include the question on how the parameter a influences the shape of the differential equation satisfied by \({}_{a}\mathtt{B}_{b, p, c}\). It is the aim of this paper to complement and to fill the void of earlier investigations on the Bessel function and its extensions.
The connection between the parameters a, b, and p in the representation formulae and recurrence relation for \({}_{a}\mathtt{B} _{b,p, c}\) are derived in Section 2. An important consequence is the derivation of an \((a+1)\)-order differential equation satisfied by the function \({}_{a}\mathtt{B}_{b,p, c}\). As applications, new functional inequalities for \({}_{a}\mathtt{B}_{b, p, -\alpha^{2}}\) are obtained, particularly in the case \(a=2\).
2 General representation formulations and applications
Proposition 1
Proposition 2
Proof
Remark 1
Proposition 3
We next find an \((a+1)\)-order differential equation satisfied by \({ }_{a}\mathtt{B}_{b, p, c}\) from the recurrence relations (5) and (6) (see also [18]).
Theorem 1
Proof
Remark 2
Proposition 4
Remark 3
Another integral representation is the following.
Proposition 5
Proof
Remark 4
3 Monotonicity and consequences
Investigations into the monotonicity properties of the generalized function \({}_{a}\mathtt{B}_{b,p,c}\) hinges on the following result of Biernacki and Krzyż [19].
Lemma 1
([19])
Suppose \(f(x)=\sum_{k=0}^{\infty }a_{k} x^{k}\) and \(g(x)=\sum_{k=0}^{\infty }b_{k} x^{k}\), where \(a_{k} \in \mathbb{R}\) and \(b_{k} > 0\) for all k. Further suppose that both series converge on \(\vert x \vert < r\). If the sequence \(\{a_{k}/b_{k}\}_{k\geq 0}\) is increasing (or decreasing), then the function \(x \mapsto f(x)/g(x)\) is also increasing (or decreasing) on \((0,r)\).
Evidently, the above lemma also holds true when both f and g are even functions, or both odd.
Theorem 2
- (a)
If \(q \geq p > -(b+1)/2\) and \(a \leq d\), then \(x \mapsto ( 2^{p}x^{-p}{}_{a}\mathtt{B}_{b,p,c}(x) ) / ( 2^{q}x^{-q} {}_{d}\mathtt{B}_{b,q,c}(x) ) \) is increasing on \((0, \infty )\).
- (b)
The function \(p\mapsto {}_{a}\mathtt{B}_{b,p+a,c}(x)/{} _{a}\mathtt{B}_{b,p,c}(x)\) is decreasing on \((-(b+1)/2, \infty )\) for each fixed \(x>0\).
- (c)
The function \(x\mapsto x{}_{a}\mathtt{B}_{b,p,c}'(x)/{} _{a}\mathtt{B}_{b,p,c}(x)\) is increasing on \((0, \infty )\) for each fixed \(p > -{(b+1)}/{2}\).
Proof
Theorem 3
- (a)
If \(\alpha \geq \beta >0\), then the function \(x \mapsto {}_{a}\mathcal{B}_{b,p,c}(x)/{}_{1}\Phi_{1}(\alpha ; \beta ; -c x^{2}/4)\) is decreasing on \(\mathbb{R}\) for each fixed \(p > -{(b+1)}/ {2}\).
- (b)If \(0< \beta \leq (2p+b+1)/(2a)\), then the function \(x \mapsto {}_{a}\mathcal{B}_{b,p,c}(x)/F_{a}(\beta ; -c x^{2}/4)\) is decreasing on \(\mathbb{R}\) for each fixed \(p > -{(b+1)}/{2}\), where$$ F_{a}(\beta , x):= {}_{0}F_{a} \biggl( -; \beta , \beta +\frac{1}{a}, \ldots , \beta +\frac{a-1}{a}; x \biggr) . $$
Proof
Lemma 2
([20])
Theorem 4
- (a)The functiongiven by (15) is decreasing and log-convex on \((-(b+1)/2, \infty )\) for each fixed \(x>0\) and \(d>0\).$$ p \mapsto {}_{a}\mathcal{B}^{d}_{b,p, c}(x) $$
- (b)The functionis increasing on \((-(b+1)/2, \infty )\), that is, for \(q \geq p >-(b+1)/2\), the inequality$$ p \mapsto {}_{a}\mathcal{B}_{b,p+1, c}^{d}(x)/{}_{a} \mathcal{B}_{b,p, c}^{d}(x) $$holds for each fixed \(x>0\) and \(d>0\).$$ {}_{a}\mathcal{B}_{b, q+1, c}^{d}(x){}_{a} \mathcal{B}_{b,p, c}^{d}(x) \geq {}_{a} \mathcal{B}_{b, q, c}^{d}(x){}_{a}\mathcal{B}_{b,p+1, c} ^{d}(x) $$(16)
- (c)
The function \(d \mapsto {}_{a}\mathcal{B}^{d}_{b,p, c}(x)\) is log-concave on \((0, \infty )\) for each fixed \(x>0\) and \(p >(2a-b-1)/2\).
Proof
To show log-convexity of \({}_{a}\mathcal{B}_{b,p, c}^{d}\), it suffices to show that \(p \mapsto \gamma_{k}(p, d)\) is log-convex for all \(k \in \{0,1, 2, 3, \ldots \}\) and fixed \(d>0\). Then the result follows from the fact that sums of log-convex functions are also log-convex.
The results of parts (a) and (b) in Theorem 4 in the case \(d=1\) were also obtained by Baricz [1, Theorem 3, Theorem 4].
Remark 5
Remark 6
Remark 7
The next result gives a dominant function for \({}_{a}\mathtt{B}_{b,p, -\alpha^{2}}\).
Theorem 5
Proof
Theorem 6
Proof
Remark 8
4 Concluding remarks
Declarations
Acknowledgements
The first author gratefully acknowledges support from FRGS research grant 203.PMATHS.6711568, and the second author acknowledges support from USM Research University Individual grant (RUI) 1001/PMATHS/8011038.
Authors’ contributions
All authors worked in coordination. All authors carried out the proof, read and approved the final version of the 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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