- Open Access
Opial inequality in q-calculus
© The Author(s) 2018
- Received: 8 February 2018
- Accepted: 2 December 2018
- Published: 14 December 2018
In this article we give q-analogs of the Opial inequality for q-decreasing functions. Using a closed form of the restricted q-integral (see Gauchman in Comput. Math. Appl. 47:281–300, 2004), we establish a new integral inequality of the q-Opial type.
- Opial’s inequality
In 1960, Opial  established the following important integral inequality.
Integral inequalities of the form (1) have an interest in itself, and also have important applications in the theory of ordinary differential equations and boundary value problems (see [1, 2, 4]). In the years thereafter, numerous generalizations, extensions and variations of the Opial inequality have appeared (see [12, 14]). The one containing fractional derivatives is investigated as well (see [3, 5]).
In a recent paper , Yang proved the following generalization of the Opial inequality.
In the paper , Jackson defined q-integral, which in the q-calculus bears his name.
The real function f defined on \([a,b]\) is called q-increasing (q-decreasing) on \([a,b]\) if \(f(qx)\le f(x)\) (\(f(qx)\ge f(x)\)) for \(x, qx \in[a,b]\). It is easy to see that if the function f is increasing (decreasing), then it is q-increasing (q-decreasing) too.
Our main results are contained in three theorems.
The following theorems are concerned with q-monotonic functions.
In this paper we have established a new general Opial type integral inequality in q-calculus. Further, we investigated the Opial inequalities in q-calculus involving two functions and their first order derivatives. We also discussed several particular cases. The method we used to establish our results is quite elementary and based on some simple observations and applications of some fundamental inequalities.
The authors are thankful to the editor and anonymous referees for their helpful comments and suggestion.
This research was done without any support.
The authors contributed equally to this work. All authors have read and approved the manuscript.
The authors declare that they have no competing interests.
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.
- Agarwal, R., Lakshmikantham, V.: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Sci, Singapore (1993) View ArticleGoogle Scholar
- Agarwal, R., Pang, P.: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Acad. Publ., Dordrecht (1995) View ArticleGoogle Scholar
- Anastassiou, G.: Balanced Canavati type fractional Opial inequalities. J. Appl. Funct. Anal. 9(1/2), 230–238 (2014) MathSciNetMATHGoogle Scholar
- Bainov, D., Simeonov, P.: Integral Inequalities and Applications. Kluwer Acad. Publ., Dordrecht (1992) View ArticleGoogle Scholar
- Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015) Google Scholar
- Gauchman, H.: Integral inequalities in q-calculus. Comput. Math. Appl. 47, 281–300 (2004) MathSciNetView ArticleGoogle Scholar
- Jackson, M.: On a q-definite integrals. Quart. J. Pure and Appl. Math. 41, 193–203 (1910) MATHGoogle Scholar
- Kac, V., Cheung, P.: Quantum Calculus. Springer, New York (2002) View ArticleGoogle Scholar
- Marinković, S., Rajković, P., Stanković, M.: The inequalities for some types of q-integrals. Comput. Math. Appl. 56, 2490–2498 (2008) MathSciNetView ArticleGoogle Scholar
- Opial, Z.: Sur une inegalite. Ann. Pol. Math. 8, 29–32 (1960) MathSciNetView ArticleGoogle Scholar
- Rajković, P., Marinković, S., Stanković, M.: Diferencijalno-integralni račun bazičnih hipergeometrijskih funkcija. Mašinski fakultet Niš (2008) Google Scholar
- Shum, D.: On a class of new inequalities. Trans. Am. Math. Soc. 204, 299–341 (1975) MathSciNetView ArticleGoogle Scholar
- Tariboon, J., Ntouyas, S.: Quantum integral inequalities on finite intervals. J. Inequal. Appl. 2014, 121 (2014) MathSciNetView ArticleGoogle Scholar
- Yang, G.: On a certain result of Z. Opial. Proc. Jpn. Acad. 42, 78–83 (1966) MathSciNetView ArticleGoogle Scholar