Enlarged integral inequalities through recent fractional generalized operators

This paper is devoted to proving some new fractional inequalities via recent generalized fractional operators. These inequalities are in the Hermite–Hadamard and Minkowski settings. Many previously documented inequalities may clearly be deduced as specific examples from our findings. Moreover, we give some comparative remarks to show the advantage and novelty of the obtained results.


Introduction
Fractional integral inequalities are very important in theoretical mathematics and are a substantial tool in dealing with fractional calculus science, which plays a vital role in modeling procedures for a variety of engineering issues [1,14,18,24,32]. Many fractional models yield better outcomes than identical equivalent models with integer derivatives, as illustrated in [27]. This drives the need for more exact inequalities when working with fractional calculus-based mathematical models. In the existing modification of a certain study, we concentrate on the most prominent Hermite-Hadamard-type inequality [2,8]. Because of the nature of its definition, convexity is crucial in analyzing inequality for convex functions; for other classes of convex functions and attributes; see [5,15,16,19,20,25,26].
Recently, generalized fractional operators have been used to construct a Hermite-Hadamard-type inequality allowing the ordinary version to be regained in its limit for the generalized fractional parameter, as shown in [1]. In [7] a generalized k-fractional integral inequality is proposed, as well as the Minkowski and Chebyshev integral inequalities that involve the generalized k-fractional integrals. Inequalities of Hermite-Hadamard type under generalized k-fractional integrals were studied in [9]. Guessab and Schmeisser [6] examined the sharp integral inequalities of the Hermite-Hadamard type. Also, Nisar et al.
Let us start with the traditional Hermite-Hadamard inequality: If u : B ⊆ R → R is a convex function and δ 1 , δ 2 ∈ B with δ 1 < δ 2 ,then Additional generalizations and expansions can be found, for instance, in [21,23,28,30]. Moreover, we can begin by recalling some basic fractional notions. and where is the gamma function.
Hyder and Barakat [10] enhanced the fractional obedient integral operators and offered more general definitions of the fractional integral operators as follows.

Definition 1.3
The general improved fractional left and right integral operators of a function H are respectively given by and where and ϑ : R + × (0, 1] → R + is a continuous function fulfilling the conditions: In 2020, Hyder and Soliman [11] introduced the new generalized theta-obedient integral where τ = q ∈ R, and θ q : R + × (0, 1] → R is a continuous function satisfying the following conditions: Using the Cauchy formula for iterated integrals, we can iterate the integral (9) n times and obtain the following result: where Replacing the natural number n by a complex number λ, we define the generalized fractional theta-obedient integral as follows.

Definition 1.4 The generalized fractional theta-obedient integral of a function H is defined by
where λ ∈ C with Re(λ) > 0, and g q is defined by (11).
In this paper, we employ recently developed generalized fractional operators to construct novel fractional inequalities for integrable nonnegative functions. These inequalities concern the Hermite-Hadamard and Minkowski inequalities. Our outcomes can be compared by the previous results established in [3,12,22]. The inequalities obtained in these references can be derived as particular cases. Also, we show in this work that the inequality of [22, Theorem 2.5] is incorrect. Finally, this paper is organized as follows: Sect. 2 contains the main results, and Sect. 3 provides concluding remarks.

Main results
In this section, we establish generalized fractional inequalities in the Hermite-Hadamard and Minkowski settings using newly discovered fractional integral operators. To support this claim, we offer the following theorems. γ )), then we have the following inequality: Proof According to the condition H(τ ) Therefore we have Using (12), we can integrate inequality (15) from 0 to t with respect to τ : Now, according to the condition H(τ ) and Using (14), we can integrate inequality (19) from 0 to t with respect to τ : Thus by adding inequalities (17) and (20) we obtain the required inequality (13).

Comparative cases
In this study, we establish the Hermite-Hadamard and Minkowski inequalities in the context of newly generalized fractional integral operators. We examine some specific cases arising from our findings in this section by presenting some particular examples of our fractional integral operator described by equation (12) as the following cases.
Case II. If we put γ = 1 and g(t, τ , γ ) = ln t -ln τ in Corollary 3.2, then we obtain the integral inequalities due to Hadamard fractional integral operator as follows.