Some k-fractional extension of Grüss-type inequalities via generalized Hilfer–Katugampola derivative

In this paper, we prove several inequalities of the Grüss type involving generalized k-fractional Hilfer–Katugampola derivative. In 1935, Grüss demonstrated a fascinating integral inequality, which gives approximation for the product of two functions. For these functions, we develop some new fractional integral inequalities. Our results with this new derivative operator are capable of evaluating several mathematical problems relevant to practical applications.


Introduction
In many problems, fractional derivatives accomplish a vital role. Fractional derivatives are used to solve many imperative real-world problems. In recent decades, this field has been highly considered by scientists and mathematicians. Fractional calculus is an important branch of applied mathematics that tackles derivatives and integrals of arbitrary orders. Fractional integral inequalities have demonstrated being one of the most significant and effective tools for the advancement of many areas of pure and applied mathematics. The latest formulations vary in various components from the existing ones. For example, classic partial derivatives are thus defined so that the classical derivatives in the sense of Newton and Leibniz are recovered within the limit, where the derivative order is an integer.
In recent years the inequalities involving fractional calculus play a very important role in all mathematical fields, which gave rise to important theories in mathematics, engineering, physics, and other fields of science. A remarkably large number of inequalities of the above type involving the special fractional integral (such as the Liouville, Riemann-Liouville, Erdelyi-Kober, Katugampola, Hadamard, and Weyl types) have been investigated by many researchers and received considerable attention: see Kiblas et al. [10].
Let , : [a, b] → R, be integrable functions such that Grüss-type inequality is defined as where the constant 1 4 is the best value, not replaceable by any other value. The paper is organized as follows. In Sect. 1, we give an introduction of the Grüss-type inequalities. In Sect. 2, we present the definition of the k-fractional integrals in the sense of Riemann-Liouville fractional integral and spaces needed for our research. In Sect. 3, we show the Grüss inequality by using the generalized k-fractional Hilfer-Katugampola derivative with the k-Rieman-Liouville integral operator. In Sect. 4, we show another inequality by using the generalized k-fractional Hilfer-Katugampola derivative with the k-Rieman-Liouville integral operator. By means of the given Grüss-type inequality we prove other inequalities. Concluding marks are given in Sect. 5.

Preliminaries
Firstly, we include some mandatory definitions and mathematical preliminaries of the fractional operators of calculus.
In case q = 1, we have M(a, b) = M q (a, b).
The generalized k-fractional Hilfer-Katugampola derivatives (left-sided and right-sided) are defined as where is the integral from Definition 2.3.

Auxiliary results
In this section, we prove a Grüss-type inequality by using the generalized k-fractional Hilfer-Katugampola derivative.
Talking the γ th derivative of this inequality with respect to y, we obtain γ -ω κ -1 y ρ-1 and integrating with respect to y from a to z, we get By (2.9) we have Again taking the γ th derivative of (3.4) with respect to ζ , we obtain which is the desired inequality.
which converts to inequality for the generalized k-Riemann-Liouville integral. . . .
Corollary 3.5 Further, if we take γ = 0 in (3.6), then we get where is the generalized k-Riemann-Liouville integral.
Then we have the following inequalities for the generalized k-fractional Hilfer-Katugampola derivative:

Corollary 3.6
Let γ = 0. The inequalities in Theorem 3.2 lead to the inequalities for the generalized k-Riemann-Liouville integral:

Corollary 3.7 Let nz
, which by Theorem 3.2 lead to the inequalities

Other related integral inequalities via generalized k-fractional Hilfer-Katugampola derivative
In this section, we prove other related integral inequalities by using the generalized kfractional Hilfer-Katugampola derivative.
Proof By Young's inequality we have Now, letting a = (y) (ζ ) and b = (ζ ) (y), we get Taking the γ th derivative with respect to y of inequality (4.5), we have Multiplying by γ -ω κ -1 y ρ-1 and integrating with respect to y from a to z, we get 1 p Applying the definition in (2.9), we obtain It follows that Again taking the γ th derivative of this inequality and then multiplying by γ -δ κ -1 ζ ρ-1 and integrating with respect to ζ from a to z, we obtain Which is the desired inequality. Now to prove the other inequalities.
To prove inequality (4.2), we follow the same steps as in the proof of inequality (4.1) by letting Similarly, the suppositions  The inequalities convert to the generalized k-fractional Riemann-Liouville integral.