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

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


Introduction
The classical Hölder's and Minkowski's inequalities are usually defined as follows.
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'sintegral. As applications, we derive some inequalities involving the incomplete -Gamma function. Let us begin with the following auxiliary results.

Auxiliary Results
We begin with the following inequality which is well known as Young's inequality.
Inequality (5) can be generalized as follows.
Inequality (6) can be written in the following form which is known as the weighted AM-GM inequality.

Some Applications to the Incomplete -Gamma Function
In this section, we derive some inequalities involving the incomplete -Gamma function. We shall use the notations Moreover, where 1 ( , ) is the -exponential integral [8].
Remark 12. The functions ( , ) and Γ ( , ) 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 .

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.