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 asfor sequences , andfor continuous function and on .
Definition 2. Let . Then the discrete and integral forms of Minkowski’s inequality are given asfor sequences , andfor 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 inequalityis satisfied.
Inequality (5) can be generalized as follows.
Lemma 4. For , let and such that . Then the inequalityis 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 inequalityis valid.
Lemma 6 (generalized Hölder’s inequality for sums). Let and such that the sums exist. Then the inequalityis 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 inequalityis 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 asprovided 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 inequalityholds 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 inequalityholds 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 asand 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 obtainwhere .
Theorem 13. For , let , , , and . Then the inequalityholds 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 obtainwhich 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 inequalityholds for .
Proof. Let in (30). Then, by the AM-GM inequality, we have
Theorem 18. For , let . Then the inequalityholds 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 inequalityis satisfied for .
Proof. By (26) we obtain which results from the weighted AM-GM inequality (7). HenceThen, 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.