- Research
- Open Access
Inequalities involving hypergeometric and related functions
- Siegfried H. Lehnigk^{1}Email author
https://doi.org/10.1186/s13660-018-1842-4
© The Author(s) 2018
- Received: 21 August 2017
- Accepted: 6 September 2018
- Published: 21 September 2018
Abstract
An inequality is being proved which is connected to cost-effective numerical density estimation of the hyper-gamma probability distribution. The left-hand side of the inequality is a combination of two in the third parameter distinct versions of the hypergeometric function at the point one. All three parameters are functions of the distribution’s terminal shape. The first and second are equal. The distinct third parameters of the two hypergeometric functions depend on terminal and initial shape. The other side of the inequality is determined by the quotient of two infinite series, which are related to the first derivatives with respect to terminal shape of the hypergeometric functions which appear in its left-hand side.
Keywords
- Hypergeometric functions
- Gamma functions
- Inequalities
- Applied statistics
- Computational problems in statistics
MSC
- 33C05
- 33B20
- 26D10
- 97K80
- 65C60
1 Introduction
For the sake of simplicity we shall from now on omit the arguments β and x whenever there is no chance for confusion, keeping in mind, however, that β in the actual independent variable, and that x is a parameter. If we attach an argument to a symbol of a dependent variable, it will be a particular value of β. For example, \(A(1) = A(1;x)\). Furthermore, derivatives will always be with respect to β and will be denoted by a prime.
2 Approximating sequences for \(Q/S\) and \(T/S\)
The facts established so far show that \(\{ F_{n} \}\) is a positive, increasing, bounded above sequence that converges for every \(\beta \in (0, \infty)\) and any fixed \(x \in (1, \infty)\) as \(n \uparrow \infty \). We want to show now that it converges to \(Q/S\) uniformly on every subinterval \([a, b]\) of \((0, \infty)\).
3 Monotonicity of \(Q/S\) and \(T/S\)
The functions \(Q/S\) and \(T/S\) are strictly monotonically increasing for \(\beta \in (0, \infty)\). It is sufficient to show this for \(Q/S\) since, by (1.10a), \(T/S = (3Q/S-1)/2\).
We now remember the fact that the polynomials \(\rho_{\nu }\) are Hurwitzian and that, consequently, the constituent terms of the polynomials \(r_{n}\) and \(p_{n}\) in (2.5) are Hurwitzian. By Theorem IV of [4] (in conjunction with the specification of terminology concerning circular regions and circles on p. 164 of [4]) the sum of any two of these constituent polynomials of degree \(4(n-1)\) is Hurwitzian. Thus, \(r_{n}\) and \(p_{n}\) are Hurwitz polynomials. Their zeros are located in the open left-hand half of the complex β-plane, which we denote by L. By another theorem [5, p. 115], all zeros of \(f'_{n}\) (and all its poles) are located in L. In other words, \(f'_{n} > 0\) for real \(\beta > 0\), a fact which has been established earlier already by direct means. Applying the theorem of [5] again, this time to the rational function \(f'_{n}\), we arrive at the result that \(f''_{n}\) has all its zeros (and poles) in L, i.e., \(f''_{n} \neq 0\) for real \(\beta > 0\). Limit relation (3.7) shows that \(f'_{n}\) decreases somewhere in the interval \((0, \infty)\). Thus, the β-derivative \(f''_{n}\) of \(f'_{n}\) must be negative somewhere. Since \(f''_{n} \neq 0\) for \(\beta > 0\), if follows that \(f''_{n} < 0\) for all \(\beta > 0\). In other words, \(f_{n}\), defined by (3.1) is concave from below on \(0 < \beta < \infty \), i.e., the tangent at any point \((\beta_{0}, f_{n}(\beta_{0}))\), \(\beta_{0} > 0\), lies above the graph of \(f_{n}\) for every \(\beta > 0\), \(\beta \neq \beta_{0}\).
4 Properties of σ
For \(\beta \in (0,1)\) we have \(0 < B < B(1) = B_{1} (\kappa (1))\), and the function \(g(B)\) in (4.10) is positive, i.e., the inequality sign in (4.10) is reversed. This proves (4.8) for the entire interval \(0 < \beta < 1\).
5 The inequality \(\sigma < Q/S\)
We turn now to our main objective and prove the inequality \(\sigma < Q/S\), \(\beta \in (0, \infty)\), fixed \(x \in (1, \infty)\). We know that \(\sigma < Q/S\) holds for small positive values of β since, by (4.2), \(\sigma \downarrow 0\) as \(\beta \downarrow 0\), and \(Q/S > x/(x+2)\) as \(\beta \downarrow 0\) by (2.15). Furthermore, \(\sigma < Q/S\) as \(\beta \uparrow \infty \) since σ decreases monotonically toward \(\sigma_{\infty }= (x-1)/(x+1) < 1\) as \(\beta \uparrow \infty \) by (4.2), whereas \(Q/S \uparrow 1\) by (2.14). (The fact that \(\sigma_{\infty }< 1\) also follows directly from \(\sigma < A < A_{ \infty }< 1\) for \(\beta \in (0, \infty)\).) Therefore, if the desired inequality \(\sigma < Q/S\) should not hold throughout, it must be violated somewhere in the interval \((0, \infty)\).
6 Declarations
6.1 Results and discussions
Inequalities have been proved which involve various combinations of psi- and hypergeometric functions. They add to the wealth of knowledge in the theory of these special function classes of higher analysis.
6.2 Conclusions
The main inequality of this paper guarantees uniqueness of the hyper-gamma parameter estimation and its application. Usefulness of this approach has been demonstrated in [6].
6.3 Methods/experimental
The aim of the study is to prove an inequality made up of functions of higher mathematical analysis. This inequality guarantees monotonicity of the first moment equation function of the four-parameter hyper-gamma probability density estimation problem. Monotonicity guarantees uniqueness of the numerical solution process. Standard analytical methods of higher analysis have been employed to accomplish the proof.
Declarations
Acknowledgements
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Availability of data and materials
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Funding
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Author’s contributions
Not applicable. Author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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Authors’ Affiliations
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