Quantum integral inequalities on finite intervals
Journal of Inequalities and Applications volume 2014, Article number: 121 (2014)
In this paper, some of the most important integral inequalities of analysis are extended to quantum calculus. These include the Hölder, Hermite-Hadamard, trapezoid, Ostrowski, Cauchy-Bunyakovsky-Schwarz, Grüss, and Grüss-Čebyšev integral inequalities. The analysis relies on the notions of q-derivative and q-integral on finite intervals introduced by the authors in (Tariboon and Ntouyas in Adv. Differ. Equ. 2013:282, 2013).
MSC: 34A08, 26D10, 26D15.
The integral inequalities play a fundamental role in the theory of differential equations. The study of the fractional q-integral inequalities is also of great importance. Integral inequalities have been studied extensively by several researchers either in classical analysis or in the quantum one; see [1–6] and references cited therein.
The purpose of this paper is to find q-calculus analogs of some classical integral inequalities. In particular, we will find q-generalizations of the Hölder, Hermite-Hadamard, trapezoid, Ostrowski, Cauchy-Bunyakovsky-Schwarz, Grüss, and Grüss-Čebyšev integral inequalities.
The paper is organized as follows: In Section 2, we shall introduce some definitions and auxiliary results which will help us to prove our main results. In Section 3, we establish our main results.
To the best of our knowledge, this paper is the first one that focuses on quantum integral inequalities on finite intervals.
2 Preliminaries and auxiliary results
Let be an interval and be a constant. We define q-derivative of a function at a point on as follows.
Definition 2.1 Assume is a continuous function and let . Then the expression
is called the q-derivative on J of function f at x.
We say that f is q-differentiable on J provided exists for all . Note that if in (2.1), then , where is the well-known q-derivative of the function defined by
For more details, see .
In addition, we should define the higher q-derivative of functions on J.
Definition 2.2 Let is a continuous function. We define the second-order q-derivative on interval J, which denoted as , provided is q-differentiable on J with . Similarly, we define higher order q-derivative on J, .
Example 2.1 Let and . Then, for , we have
For , we have .
Lemma 2.1 
Let , then we have
The q-integral on interval J is defined as follows.
Definition 2.3 Assume is a continuous function. Then the q-integral on J is defined by
for . Moreover, if then the definite q-integral on J is defined by
Note that if , then (2.4) reduces to the classical q-integral of a function , defined by for . For more details, see .
Example 2.2 Let for , then we have
Example 2.3 Let a constant , then we have
Note that if , then (2.5) reduces to the classical integration
Theorem 2.1 
Let be a continuous function. Then we have:
Theorem 2.2 
Assume are continuous functions, . Then, for ,
For the basic properties of q-derivative and q-integral on finite intervals, we refer to .
Lemma 2.2 For , the following formula holds:
Proof Let , and , then, by Definition 2.1, we have
Applying q-integral on J for (2.7), we obtain (2.6) as required. □
Example 2.4 Let and . Then, from q-integrating by parts and Lemmas 2.1 and 2.2, we have
3 Quantum integral inequalities on finite intervals
In this section, some of the most important integral inequalities of analysis are extended to quantum calculus. We start with the q-Hölder inequality on the interval .
Theorem 3.1 Let , , such that . Then we have
Proof From Definition 2.3 and the discrete Hölder inequality, we have
Therefore, inequality (3.1) is valid. □
Remark 3.1 If , then inequality (3.1) reduces to the classical q-Hölder inequality in [, p.604].
Next, we present the q-Hermite-Hadamard integral inequality on .
Theorem 3.2 Let be a convex continuous function on J and . Then we have
Proof The convexity of f on means that
for all .
Taking q-integration for (3.3) over t on , we have
From Example 2.2, we have
Definition of q-integration on J leads to
which gives the second part of (3.2) by using (3.4).
To prove the first part of (3.2), we use the convex property of f as follows:
Again q-integrating to the above inequality over t on and changing variables, we get
The proof is completed. □
Remark 3.2 If , then inequality (3.2) reduces to the Hermite-Hadamard integral inequality
Next comes the q-trapezoid inequality on the interval . We use the notation for the usual supremum norm on .
Theorem 3.3 Let be a q-differentiable function with continuous on and . Then we have
Proof The q-integration by parts on interval J gives
Using the properties of modulus for (3.6), we obtain
Applying Examples 2.2 and 2.3, we have
Combining (3.7) and (3.8), we obtain inequality (3.5) as required. □
Remark 3.3 If , then inequality (3.5) reduces to the well-known trapezoid inequality as
The next theorem deals with the q-trapezoid inequality with second-order q-derivative on .
Theorem 3.4 Let be a twice q-differentiable function with continuous on and . Then we have
Proof The q-integration by parts on interval J two-times and taking into account Example 2.1, we have
from Lemma 2.2 and Example 2.4, we have
Combining (3.10) and (3.11), we deduce that inequality (3.9) is valid. □
Remark 3.4 If , then inequality (3.9) reduces to the trapezoid inequality in terms of the second derivative as
In the following theorem we establish the q-Ostrowski integral inequality on interval J.
Theorem 3.5 Let be a q-differentiable function with continuous on and . Then we have
Proof Applying the Lagrangian mean value theorem , for , it follows that
Taking into account Examples 2.2 and 2.3, for , we obtain
The inequality (3.12) are obtained by combining (3.13) and (3.14). □
Remark 3.5 If , then inequality (3.12) reduces to the classical Ostrowski integral inequality as
Let us prove the q-Korkine identity on interval J.
Lemma 3.1 Let be continuous functions on J and . Then we have
Proof From Definition 2.3, we have
from which one deduces (3.15). □
Now, we will prove the q-Cauchy-Bunyakovsky-Schwarz integral inequality for double integrals on .
Lemma 3.2 Let be continuous functions on J and . Then we have
Proof According to Definition 2.3, we have the double q-integral on J as
Applying the discrete Cauchy-Schwarz inequality, we have
Therefore, inequality (3.16) is valid. □
Remark 3.6 If , then Lemmas 3.1 and 3.2 are reduced to the usual Korkine identity and Cauchy-Bunyakovsky-Schwarz integral inequality for double integrals, respectively. For more details, see  and .
We define the q-Čebyšev functional on interval J by
By using Lemmas 3.1 and 3.2 coupled with (3.17), we obtain the q-Grüss integral inequality on interval . The proof of the following theorem is similar to the classical Grüss integral inequality; see [3, 9]. Therefore, we omit it.
Theorem 3.6 Let be continuous functions on and satisfy
Then we have the inequality
Remark 3.7 The inequality (3.19) is similar to q-Grüss integral inequality in . However, the results from  obtained by using the restricted definite q-integral which is a finite sum as a special type of the definite q-integral.
Now, we are going to prove the q-Grüss-Čebyšev integral inequality on interval .
Theorem 3.7 Let be -Lipschitzian continuous functions on , so that
for all . Then we have the inequality
Proof We recall the q-Korkine identity on interval J as
From (3.20), we get
for all .
The double q-integration for (3.23) on leads to
Note that if , then (3.25) reduces to the integral
By direct computation, we have
Thus, from (3.24) and (3.26), we obtain
Using (3.22), we obtain (3.21). □
Remark 3.8 If , then inequality (3.21) reduces to the classical Grüss-Čebyšev integral inequality as
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
Anastassiou GA: Intelligent Mathematics: Computational Analysis. Springer, New York; 2011.
Belarbi S, Dahmani Z: On some new fractional integral inequalities. JIPAM. J. Inequal. Pure Appl. Math. 2009., 10: Article ID 86
Cerone P, Dragomir SS: Mathematical Inequalities. CRC Press, New York; 2011.
Dahmani Z: New inequalities in fractional integrals. Int. J. Nonlinear Sci. 2010, 9: 493–497.
Dragomir SS: Some integral inequalities of Grüss type. Indian J. Pure Appl. Math. 2002, 31: 397–415.
Ogunmez H, Ozkan UM: Fractional quantum integral inequalities. J. Inequal. Appl. 2011., 2011: Article ID 787939
Kac V, Cheung P: Quantum Calculus. Springer, New York; 2002.
Tariboon J, Ntouyas SK: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013., 2013: Article ID 282
Pachpatte BG: Analytic Inequalities. Atlantis Press, Paris; 2012.
Florea A, Niculescu CP: A note on Ostrowski’s inequality. J. Inequal. Appl. 2005,2005(5):459–468.
Gauchman H: Integral inequalities in q -calculus. Comput. Math. Appl. 2004, 47: 281–300. 10.1016/S0898-1221(04)90025-9
The research of J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
The authors declare that they have no competing interests.
Both authors contributed equally in this article. They read and approved the final manuscript.
About this article
Cite this article
Tariboon, J., Ntouyas, S.K. Quantum integral inequalities on finite intervals. J Inequal Appl 2014, 121 (2014). https://doi.org/10.1186/1029-242X-2014-121
- q-integral inequalities
- Hölder’s inequality
- Hermite-Hadamard’s inequality
- Ostrowski’s inequality
- Grüss-Čebyšev integral inequality