Quantum integral inequalities on finite intervals
© Tariboon and Ntouyas; licensee Springer. 2014
Received: 27 January 2014
Accepted: 14 March 2014
Published: 26 March 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.
is called the q-derivative on J of function f at x.
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, .
For , we have .
Lemma 2.1 
The q-integral on interval J is defined as follows.
Note that if , then (2.4) reduces to the classical q-integral of a function , defined by for . For more details, see .
Theorem 2.1 
Theorem 2.2 
For the basic properties of q-derivative and q-integral on finite intervals, we refer to .
Applying q-integral on J for (2.7), we obtain (2.6) as required. □
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 .
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 .
for all .
which gives the second part of (3.2) by using (3.4).
The proof is completed. □
Next comes the q-trapezoid inequality on the interval . We use the notation for the usual supremum norm on .
Combining (3.7) and (3.8), we obtain inequality (3.5) as required. □
The next theorem deals with the q-trapezoid inequality with second-order q-derivative on .
Combining (3.10) and (3.11), we deduce that inequality (3.9) is valid. □
In the following theorem we establish the q-Ostrowski integral inequality on interval J.
The inequality (3.12) are obtained by combining (3.13) and (3.14). □
Let us prove the q-Korkine identity on interval J.
from which one deduces (3.15). □
Now, we will prove the q-Cauchy-Bunyakovsky-Schwarz integral inequality for double integrals on .
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 .
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.
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 .
for all .
Using (3.22), we obtain (3.21). □
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
The research of J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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