Integral inequalities via Raina’s fractional integrals operator with respect to a monotone function

We establish certain new fractional integral inequalities involving the Raina function for monotonicity of functions that are used with some traditional and forthright inequalities. Taking into consideration the generalized fractional integral with respect to a monotone function, we derive the Grüss and certain other associated variants by using well-known integral inequalities such as Young, Lah–Ribarič, and Jensen integral inequalities. In the concluding section, we present several special cases of fractional integral inequalities involving generalized Riemann–Liouville, k-fractional, Hadamard fractional, Katugampola fractional, (k, s)-fractional, and Riemann–Liouville-type fractional integral operators. Moreover, we also propose their pertinence with other related known outcomes.


Introduction and preliminaries
The fractional calculus has gained importance during recent years because of its applications in science and engineering. Fractional-order differential equations are widely used in the model problems of nanoscale flow and heat transfer, diffusion, polymer physics, chemical physics, biophysics, medical sciences, turbulence, electric networks, electrochemistry of corrosion, and fluid flow through porous media [1][2][3][4][5]. Fractional integral inequalities associating functions of two or more independent variables play a crucial role in the continuous growth of the theory, methods, and applications of differential and integral equations. In view of wider applications, integral inequalities have received considerable attention. Recently, several refinements of fractional integral inequalities have been proposed, which are helpful in the study of distinct classes of differential and integral equations.
It is well known that the Grüss-type inequalities in both continuous and discrete cases play a significant role in investigating the qualitative conduct of differential and difference equations, respectively, as well as several other fields of pure and applied analysis.
Getting this tendency, we present a novel version for the most aesthetic and useful Grüss-type inequality [10] and some other associated inequalities with respect to another function ϑ that could be progressively viable and, moreover, more appropriate than the previous ones. The Grüss inequality can be stated as follows.
Inequality (1.1) is a tremendous mechanism for investigating numerous scientific areas of research comprising engineering, fluid dynamics, biosciences, chaos, meteorology, vibration analysis, biochemistry, aerodynamics, and many more. There was a constant development of enthusiasm for such an area of research so as to address the issues of different utilizations of these variants [11][12][13][14][15]. The conventional theory of inequality is unable to clarify the true behavior of (1.1). A review of basic concepts of fractional integral inequalities and an understanding about the Grüss was presented by Dahmani et al. [16]. Rashid et al. [17,18] formulated the governing inequality by using generalized k-fractional integral and generalized proportional fractional integral. Based on a monotone function, Rashid et al. [19] derived fractional integral inequalities by means of the generalized proportional fractional integral operator in the sense of another function. Very recently, Butt et al. [20] proposed novel fractional refinements of Čebyšev-Pólya-Szegö-type inequalities by using the Raina function in the kernel. Now we evoke some preliminaries ideas, which help the readers in clear understanding. The Mellin transform of the exponential function e -t k k is the k-gamma function given by
The principal purpose of this paper is deriving novel identities, integral inequalities including a Grüss-type inequality, and numerous other associated inequalities via generalized fractional integral inequalities with respect to other function ϑ by using Young's, weighted arithmetic and geometric mean inequalities, and so on. It is interesting that many particular cases can be revealed by using Remarks 1.6 and 1.7. Therefore it is necessary to propose the investigation of the generalized fractional integrals.

Fractional Grüss-type inequalities
To demonstrate the main consequences of this paper, we begin with certain integral inequalities and equalities for positive integrable functions with the generalized fractional integral operator having the well-known Raina function in its kernel.
(ii) A λ and A δ are defined by respectively.

Lemma 2.2
If all the conditions of Theorem 2.1 are satisfied, then we have the equality where A λ (x) is defined by (2.1).
Proof Let φ 1 , φ 2 ∈ R, and let Q 1 be a function defined on [0, ∞). Then for any t > 0 and η > 0, we have Multiplying both sides of (2.8) by and integrating the obtained result with respect to t over (0, x) lead to Again, multiplying both sides of (2.9) by and integrating the obtained result with respect to η over (0, x) give which completes the proof of Lemma 2.2.

Lemma 2.3 Under the assumptions of Theorem 2.1, we have
where A λ (x) and A δ (x) are defined by (2.1) and (2.2), respectively.
Proof Multiplying both sides of (2.9) by and integrating the obtained results with respect to t over (0, x) lead to which completes the proof of Lemma 2.3.

Theorem 2.4 Under the assumptions of Theorem 2.1, we have
(2.14) Multiplying both sides of (2.14) by and integrating the obtained result with respect to t and η over (0, x) give the desired inequality (2.13).
Proof Let t, η ∈ [0, ∞). Then from inequality (2.15) we clearly see that which implies that Multiplying both sides of (2.17) by and integrating the obtained result with respect to t and η over (0, x) lead to the desired inequality (2.16).
Proof We first prove part (i). For x ∈ [0, ∞), from (2.18) it follows that Multiplying both sides of (2.31) by and integrating the obtained result with respect to t and η over (0, x) lead to the desired inequality in part (i).
To prove parts (ii)-(iv), we only need to use the inequalities By adopting a similar procedure as we did in the theorem we can easily derive the following lemma.
Proof The theorem can be easily proved by using the Lah-Ribarič inequality [29,30].
Proof The theorem can be proved by using the Jensen inequality for convex functions.

Concluding remarks
This section is dedicated to several particular cases of the main consequences derived in Sects. 2 and 3. I. If we choose ω = 0, λ = α, and σ (0) = 1, then under the assumptions of Theorem 2.4, we get the result for one-sided generalized k-fractional integral proposed by Rashid et al. [17].
More related results can be derived by using similar methods in Sects. 2 and 3, and we leave the details to the interested readers.

Conclusion
In the paper, we established new Grüss-type fractional integral inequalities and several other associated variants by employing the generalized fractional integral functions having the Raina function in its kernel. Furthermore, we derived numerous novel variants for the monotonicity of functions. Numerous particular cases can be discussed with consideration of Remarks 1.6 and 1.7, which we can supposed as a significant modification of the earlier consequences. For an appropriate choice of ω, λ, and σ (0) = 1, we can acquire several novelties, which need further investigations. We hope that novelties concerned with our generalizations can bring revolutionary development and also be implemented in differential and difference equations.