Abstract and Applied Analysis
Volume 2017 (2017), Article ID 9796873, 6 pages
https://doi.org/10.1155/2017/9796873

Research Article

Generalized Hölder’s and Minkowski’s Inequalities for Jackson’s -Integral and Some Applications to the Incomplete -Gamma Function

Kwara Nantomah

Department of Mathematics, University for Development Studies, Navrongo Campus, P.O. Box 24, Navrongo, Upper East Region, Ghana

Correspondence should be addressed to Kwara Nantomah; knantomah@uds.edu.gh

Received 27 March 2017; Accepted 18 June 2017; Published 16 July 2017

Academic Editor: Shanhe Wu

Copyright © 2017 Kwara Nantomah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We establish some generalized Hölder’s and Minkowski’s inequalities for Jackson’s -integral. As applications, we derive some inequalities involving the incomplete -Gamma function.

1. Introduction

The classical Hölder’s and Minkowski’s inequalities are usually defined as follows.

Definition 1. Let and . Then the discrete and integral forms of Hölder’s inequality are given as

for sequences , and

for continuous function and on .

Definition 2. Let . Then the discrete and integral forms of Minkowski’s inequality are given as

for sequences , and

for continuous function and on .

These fundamental results are well known in the literature and have been studied intensively by several researchers. Their role in mathematics and related disciplines is invaluable. For instance, they play a pivotal role in classical real and complex analysis, probability theory, statistics, numerical analysis, and so on. Over the past years, various refinements, extensions, and applications have appeared in the literature. In the present work, our objective is to provide some generalized Hölder’s and Minkowski’s inequalities for Jackson’s -integral. As applications, we derive some inequalities involving the incomplete -Gamma function. Let us begin with the following auxiliary results.

2. Auxiliary Results

We begin with the following inequality which is well known as Young’s inequality.

Lemma 3. For , , and , the inequality

is satisfied.

Inequality (5) can be generalized as follows.

Lemma 4. For , let and such that . Then the inequality

is valid.

Inequality (6) can be written in the following form which is known as the weighted AM-GM inequality.

Lemma 5. For , let and such that . Then the inequality

is valid.

Lemma 6 (generalized Hölder’s inequality for sums). Let and such that the sums exist. Then the inequality

is valid for such that .

Proof. The inequality is obvious if for each . So we assume and let

Then by the generalized Young’s inequality (6), we obtain

By adding these inequalities, we obtain

which gives inequality (8).

Lemma 7 (generalized Minkowski’s inequality for sums). Let and such that the sums exist. Then the inequality

is valid for .

Proof. We prove this by mathematical induction on . If , then (12) reduces to the classical Minkowski’s inequality (3). Thus (12) is valid for . Next, assume that (12) holds for some . Based on this assumption, we want to show that (12) holds for . We proceed as follows:

Thus (12) holds for and this completes the proof.

Remark 8. Inequalities (8) and (12) are already known in the literature. See, for instance, [1–3]. Here we offer simple proofs of the results.

3. Generalized -Hölder’s and -Minkowski’s Inequalities

Jackson’s -integral from to and that from to are, respectively, defined as

provided that the sums in (14) and (15) converge absolutely [4]. In a generic interval , the -integral takes the following form:

Theorem 9 (generalized -Hölder’s inequality). Let be functions such that the integrals exist. Then the inequality

holds for such that .

Proof. Let such that . Then by relation (14) and inequality (8), we obtain

which concludes the proof.

Remark 10. Let , , , , and in Theorem 9. Then we obtain the result of Lemma  2.1 of [5].

Theorem 11 (generalized -Minkowski’s inequality). Let be functions such that the integrals exist. Then the inequality

holds for .

Proof. Similarly we apply the principle of mathematical induction. For , inequality (19) reduces to -Minkowski’s inequality obtained in Remark  4 of [6]. Assume that (19) holds for . Based on this assumption, we show that (19) holds for . That is,

Thus (19) holds for . This completes the proof.

4. Some Applications to the Incomplete -Gamma Function

In this section, we derive some inequalities involving the incomplete -Gamma function. We shall use the notations and subsequently.

El-Shahed and Salem [7] defined the incomplete -Gamma function for as

and the complementary incomplete -Gamma function as

where is a -analogue of the classical exponential function, , and . One can easily see that

where is the -Gamma function. Also, the following identities are satisfied:

Moreover,

where is the -exponential integral [8].

Remark 12. The functions and can be viewed as both functions of (for fixed ) and functions of (for fixed ). For the purpose of this paper, we shall concentrate on as functions of .

By differentiating (21) times, we obtain

where .

Theorem 13. For , let , , , and . Then the inequality

holds for .

Proof. By (26) and the generalized -Hölder’s inequality (17), we obtain

which completes the proof.

Remark 14. Let , , , , and in Theorem 13. Then, we obtain

Remark 15. Let in (29). Then we obtain

which implies that the function is logarithmically convex. Also, since , then it follows that is also logarithmically convex.

Remark 16. Let and in (29). Then we obtain the Turan-type inequality

Corollary 17. Let . Then the inequality

holds for .

Proof. Let in (30). Then, by the AM-GM inequality, we have

Theorem 18. For , let . Then the inequality

holds for and .

Proof. Note that for and . Then by the generalized -Minkowski’s inequality (19), we obtain

which completes the proof.

Remark 19. In particular, let , , , , and in Theorem 18. Then we obtain

Theorem 20. For , let , , , and . Then the inequality

is satisfied for .

Proof. By (26) we obtain

which results from the weighted AM-GM inequality (7). Hence

Then, by exponentiating (39), we obtain the required result (37).

Remark 21. Results of types (27), (34), and (37) which deal with the -Gamma function can also be found in [9].

5. Conclusion

In this study, we provided simple proofs of the discrete forms of some generalized Hölder’s and Minkowski’s inequalities. Based on these results, we established some generalized Hölder’s and Minkowski’s inequalities for Jackson’s -integral. Furthermore, by using the established results, we derived some new inequalities involving the incomplete -Gamma function. We anticipate that the present results will find some applications in -Calculus as well as other related disciplines.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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