A Generalized -Grüss Inequality Involving the Riemann-Liouville Fractional -Integrals
The aim of this paper is to establish -extension of the Grüss type integral inequality related to the integrable functions whose bounds are four integrable functions, involving Riemann-Liouville fractional -integral operators. The results given earlier by Zhu et al. (2012) and Tariboon et al. (2014) follow the special cases of our findings.
Let and be two continuous functions defined on , such that and , for each , where , , , and are given real constants; then In the literature several generalizations of the Grüss type integral inequality are considered by many researchers (see [3–10]). Dahmani et al.  established a generalization of inequality (1), using Riemann-Liouville fractional integrals, as follows.
Let and be two integrable functions with constant bounds defined on , such that then for where denote the Riemann-Liouville fractional integral operator of order for the function . Recently, by replacing the constants appearing as bounds of the functions and by four integrable functions, Tariboon et al.  investigate more general forms of inequality (3).
The subject of -calculus has gained noticeable importance due to applications in mathematics, statistics, and physics. Particularly, the -analysis has found many applications in the theory of partitions, combinatorics, exactly solvable models in statistical mechanics, computer algebra, geometric functions theory, optimal control problems, -difference, and -integral equations [13–16]. This has led various workers in the field of -theory for exploring the possible -extensions to all the important results available in the classical theory. With this objective in mind, Gauchman  investigated -analogues of some classical integral inequalities, including the well-known Grüss inequality (1). Further, a number of authors have studied, in depth, the -extension and applications of various classical integral inequalities (see [18–23]). Very recently, Zhu et al.  derived -extension of inequality (3) and certain new fractional -integral inequalities on the specific time scale.
It is fairly well-known that there are a number of different definitions of fractional integrals and their applications. Each definition has its own advantages and suitable for applications to different type of problems. All specialists of this field (Fractional Calculus) know the importance of different types of definitions of fractional calculus operators and their use in specified problems. It is to be noted that the problem considered here provides unifications to the results of [12, 24] and gives a generalized -Grüss type integral inequality related to the integrable functions whose bounds are four integrable functions, involving Riemann-Liouville fractional -integrals. Additionally, Riemann-Liouville fractional -integral operator has the advantage that it generalizes the familiar Riemann-Liouville operator and also provides the results on time scales. Our main result provides -extension of the result due to Tariboon et al.  and can be further applied to derive certain interesting consequent results and special cases.
The -shifted factorial is defined for , as a product of factors by and in terms of the basic analogue of the gamma function where the -gamma function is defined by ([13, p. 16, eqn. (1.10.1)])
Further, we note that and if , definition (4) remains meaningful for , as a convergent infinite product given by
Also, the -binomial expansion is given by Jackson’s -derivative and -integral of a function defined on are, respectively, given by (see [13, pp. 19, 22]) The Riemann-Liouville fractional -integral operator of a function of order (due to Agarwal ) is given by where
2. A Generalized -Grüss Integral Inequality
Our results in this section are based on the following lemma, giving functional relation for Riemann-Liouville fractional -integral operators, with the integrable functions.
Lemma 1. Let , , and be integrable functions defined on , such that Then, for and , we have
Proof. On using the hypothesis of inequality (14), for any , we can write Consider for all . Multiplying both sides of (16) by and integrating the resulting identity with respect to from to , and using integral operator (11), we get Next, on multiplying both sides of (18) by , where is given by (17), and integrating with respect to from to , we obtain which upon using formula (13) (for ), we easily arrive at the identity (15).
Now, we obtain a generalized -Grüss integral inequality, which gives an estimation for the fractional -integral of a product in terms of the product of the individual function fractional -integrals, involving Riemann-Liouville fractional hypergeometric operators. Our inequality is related to the integrable functions and , whose bounds are integrable functions and satisfying the Cauchy-Schwarz inequality.
Theorem 2. Let and be two integrable functions on and , , , and are four integrable functions on , such that Then, for and , one has where
Proof. Let and be two integrable functions on and satisfying inequality (20); then for any , we define a function
On multiplying both sides of (23) by , where and are given by (17), and integrating with respect to and , respectively, from to , we obtain
Now, upon using the Cauchy-Schwarz inequality for -integrals (for details, see ), we get
On the other hand, we observe that each term of the series in (17) is positive, and hence, the function remains positive, for all . Therefore, under the hypothesis of Lemma 1, it is obvious to see that either if a function is integrable and nonnegative on , then or if a function is integrable and nonpositive on , then .
Now, by noting the relation that, for all , we have Thus, upon using Lemma 1, we get Similarly, we can write On making use of inequalities (25), (28), and (29), we easily arrive at the main result (21).
Now, we briefly consider some special cases of the result derived in the preceding section. If we let , and we make use of the limit formulas: we observe that inequality (21) of Theorem 2 provides the -extension of the known result due to Tariboon et al. [12, p. 5, Theorem 9].
Corollary 3. Let and be two integrable functions on , such that
Again, if we set , , , and , then Theorem 2 leads to the following -integral inequality.
Corollary 4. Let and be two integrable functions on , such that Then, for and , one has where
We conclude this paper by remarking that we have introduced a new general extension of -Grüss type integral inequality, which gives an estimation for the fractional -integral of a product in terms of the product of the individual function fractional -integrals involving Riemann-Liouville fractional integral operators. Our main result is related to the integrable functions and , whose bounds are integrable functions. Therefore, by suitably specializing the arbitrary functions , , , and , one can easily investigate additional integral inequalities from our main result Theorem 2.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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