New general Grüss-type inequalities over σ-finite measure space with applications

In this paper, we establish some new integral inequalities involving general kernels. We obtain the related broad range of fractional integral inequalities. Also, we apply the Young inequality to find new forms of inequalities for generalized kernels. These new and motivated results generalize the results for fractional integrals such that fractional integral of a function with respect to an increasing function, Riemann–Lioville fractional integrals, Erdélyi–Kober fractional integrals, Hadamard fractional integrals, generalized factional integral integrals in addition to the corresponding k-fractional integrals.


Introduction
Fractional calculus deals with the study of derivative and integral operators of fractional order. This field is as important as calculus itself. In the last few decades, it attracted many researchers, who produced remarkable work (see, e.g., [1-3, 6, 7, 9-13, 18]). In particular, the uniqueness of solutions for fractional partial differential equations can be established by using fractional integral inequalities.
The Grüss inequality connects the integral of the product of two functions with the product of their integrals. Our main purpose in this paper is showing some new modifications of the Grüss inequality by using a general kernel. The Grüss inequality is one of the most fascinating inequalities and stated in the following theorem.  (1.1) where the constant 1 4 cannot be improved.
Let ( , Σ, μ) be a measure space with positive σ -finite measure, let k : × → R be a nonnegative function, and let Θ(x) = k(x, y) dμ(y), x ∈ . (1.2) Throughout this paper, we suppose Θ(x) > 0 a.e. on . Let U(k) denote the class of functions Λ : → R with the representation where Λ : → R is a measurable function. and where Γ is the gamma function.
Diaz et al. [4] defined the gamma k-function as follows.
be a finite or infinite interval of the real line , and let α > 0. Let g be an increasing and positive monotone function on (a, b]. The left-and right-sided fractional integrals of a function f with respect to g in [a, b] are given by A more general form of Definition 1.5 is as follows. Definition 1.6 Let k > 0, let (a, b), -∞ ≤ a < b ≤ ∞, be a finite or infinite interval of the real line , and let α > 0. Let g be an increasing and positive monotone function on (a, b]. The left-and right-sided fractional integrals of a function f with respect to g of order α, k > 0 in [a, b] are given by and Now we continue with the definition of Hadamard-type fractional integrals. Definition 1.7 Let (a, b) be a finite or infinite interval of the half-axis + , and and α > 0. The left-sided and right-sided Hadamard-type fractional integrals of order α > 0 are given by respectively.
The generalized Hadamard-type fractional integrals are defined as follows.
Definition 1.8 Let (a, b) be a finite or infinite interval of R + , and let α > 0. The left-and right-sided Hadamard-type fractional integrals of order α > 0 are given by and respectively.
Note that Hadamard fractional integrals of order α are a particular case of the left-and right-sided fractional integrals of a function f with respect to the function g( Now we present the definition of the Erdélyi-Kober-type fractional integrals. Some of these definitions and results were presented in Samko et al. [16]. Definition 1.9 Let (a, b) (0 ≤ a < b ≤ ∞) be a finite or infinite interval of the half-axis R + . Let α > 0, σ > 0, and η ∈ R. We consider the left-and right-sided integrals of order α ∈ R defined by and respectively. Integrals (1.4) and (1.5) are called the Erdélyi-Kober-type fractional integrals.

Main results
The first main result is given in the following: Suppose that the exist integrable functions Ψ 1 , Ψ 2 on [0, ∞), such that Proof Using (2.1), for all γ ≥ 0 and δ ≥ 0, we have Then Multiplying both sides of (2.3) by k(ξ , γ ) and integrating the resulting identity with respect to γ over , we get which can be written as Now multiplying both sides of (2.4) by k(ξ , δ) and integrating the resulting identity with respect to δ over , we get This completes the proof.
Proof Since γ , δ > 0, we have Multiplying both sides by k(ξ , γ ) and integrating with respect to the variable γ over , we get Again multiplying both sides by k(ξ , δ) and integrating with respect to the variable δ over , we get This completes the proof.