This paper is motivated by an open problem of several inequalities for confluent hypergeometric functions. We give some inequalities for simple Laguerre, Laguerre, pseudo-Laguerre, Shively’s pseudo-Laguerre, and Hermite functions of one and two variables.
Hinweise
Competing interests
The author declares that they have no competing interests.
1 Introduction
Inequalities for the special functions appear infrequently in the literature. Some of these inequalities are closely related to those presented here. Inequalities for the ratio of confluent hypergeometric functions are available in the literature. The formulas are very important, as they include expansions for many transcendent expressions of mathematical physics in series of the classical orthogonal polynomials, Laguerre, Hermite functions, etc. [1‐8]. The developments bear heavily on the works of Luke [9‐12]. The author has earlier studied the inequalities for Humbert functions [13]. This motivated us to obtain two-sided inequalities for simple Laguerre, Laguerre, pseudo-Laguerre, Shively’s pseudo-Laguerre, and Hermite functions of one and two variables through confluent hypergeometric functions of one and two variables.
In [14], the confluent hypergeometric function is defined as
$$ {}_{1}F_{1}(a;c;x) = \sum_{k \ge 0} \frac{(a)_{k}}{k!(c)_{k}}x^{k},\quad c > 0, x > 0, $$
(1.1)
where \((a)_{k}\) is the Pochhammer symbol or shifted factorial defined as
$$(a)_{k} = a(a + 1)(a + 2)\cdots(a + k - 1) = \frac{\Gamma (a + k)}{\Gamma (a)};\quad k \ge 1;(a)_{0} = 1. $$
Anzeige
We recall here the theorems in [15, 16], which will be used in the investigation that follows.
The purpose of this section is to introduce new inequalities for a simple Laguerre function of one and two variables, which represents the hypergeometric functions as given by the relations (1.1)-(1.10).
The simple Laguerre function \(L_{a}(x)\) of one variable is defined by [14]
$$ L_{a}(x) ={}_{1}F_{1} ( - a;1;x ). $$
(2.1)
In the following some inequalities are established.
Theorem 2.1
If\(a < 0\)and\(0 < x < 1\), then we have
$$ 1 - ax < L_{a}(x) < 1 - 2ax. $$
(2.2)
Proof
Equation (2.2) follows directly from (2.1) and (1.2) by putting \(c = 1\). □
Similarly result (2.3), (2.4), and (2.5) can also be obtained with the help of (2.1), (1.3), (1.5), and (1.6). □
Our interest is to show that our inequality (2.3) gives inequality at \(x = 0.9\) and \(a = - 0.5\);
$$1.45 < L_{ - 0.5}(0.9) < 1.9. $$
Note also that not only the restriction that the argument is less than the order is waived, but also it holds in the extended domain \(a \in ( - 1,0)\), \(x = 0.5\), and \(y = 1\)
The numerical computation appended below verifies these ratios under suitable restrictions, and gives bounds of ratios otherwise readily available. For the examples
The previous properties can be generalized as follows: We define the classical Laguerre function \(L_{a,b}^{(\alpha,\beta )}(x,y)\) of two variables in the form
The proof of the theorem is very similar to Theorem 2.3. □
In particular, for the set of values \(\alpha = 1,0.5,0.7\), \(\beta = 2,0.7,0.6\), \(x = 0.9,0.8\), and \(y = 0.6,0.4\), we have from (3.8), (3.9), and (3.10), respectively, the inequalities:
In the next section, we discuss some inequalities for pseudo-Laguerre functions for different values that exhibit very interesting behavior of these function.
4 Inequalities for pseudo-Laguerre functions
For the pseudo-Laguerre function of one variable \(f_{a}(x;\lambda )\) [14, 18] defined by
In the next section, we show that there exist operational relations between confluent hypergeometric functions and the various types of Shively’s pseudo-Laguerre functions.
5 Inequalities for Shively’s pseudo-Laguerre functions
Shively [18] studied the pseudo-Laguerre function of one variable by
By using inequalities of a confluent hypergeometric function, we can obtain some inequalities for Shively’s pseudo-Laguerre function in the following theorem.
Theorem 5.1
$$\begin{aligned}& \biggl( 1 - \frac{a}{\alpha + a}x \biggr) \biggl( \frac{(\alpha )_{2a}}{\Gamma (a + 1)(\alpha )_{a}} \biggr) < R_{a}(\alpha,x) < \biggl( 1 - \frac{2a}{\alpha + a}x \biggr) \biggl( \frac{(\alpha )_{2a}}{\Gamma (a + 1)(\alpha )_{a}} \biggr); \\& \quad a < 0,\alpha + a > 0,0 < x < 1, \end{aligned}$$
(5.2)
$$\begin{aligned}& e^{x - y} < \frac{R_{a}(\alpha,x)}{R_{a}(\alpha,y)} < 1; \quad \alpha + a > - a > 0, y > x > 0, \end{aligned}$$
Using the inequalities of confluent hypergeometric function (1.7)-(1.10) and (5.5), we obtain the inequalities for Shively’s pseudo-Laguerre function. □
As a numerical verification, we have from Theorem 5.2:
Similarly the results (6.17)-(6.19) can also be obtained with the help of (1.7)-(1.10) and (6.16). □
In conclusion, Luke’s technique is to a large extent successful in obtaining inequalities for the confluent hypergeometric function. The theorems developed herein seem sufficient to indicate the general nature of expected results and we do not further pursue the subject. Here, we have obtained inequalities for simple Laguerre, Laguerre, pseudo-Laguerre, Shively’s pseudo-Laguerre, and Hermite functions from a well-known result for confluent hypergeometric functions and their confluent cases. In a future work, we intend to investigate this aspect of the subject and to apply our techniques to develop inequalities for hypergeometric functions of several variables.
Acknowledgements
The author expresses sincere appreciation to Dr. Mohamed Saleh Metwally (Department of Mathematics, Faculty of Science (Suez), Suez Canal University, Egypt), and Dr. Mahmoud Tawfik Mohamed (Department of Mathematics, Faculty of Science (New Valley), Assiut University, New Valley, EL-Kharga 72111, Egypt) for their kind interests, encouragement, help, suggestions, comments and the investigations for this series of papers. The author would like to thank the anonymous referees for valuable comments and suggestions, which have led to a better presentation of the paper.
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.
Competing interests
The author declares that they have no competing interests.