Some unified bounds for exponentially tgs-convex functions governed by conformable fractional operators

1 School of Science, Huzhou University, Huzhou 313000, P. R. China 2 Department of Mathematics, Government College University, Faisalabad 38000, Pakistan 3 Department of Mathematics, COMSATS University, Islamabad 44000, Pakistan 4 Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China 5 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China


Introduction
On different time ranges, fractional calculus has a great impact due to a diversity of applications that have contributed to several fields of technical sciences and engineering [1][2][3][4][5][6][7][8][9][10][11][12]. One of the principal options behind the popularity of the area is that fractional-order differentiations and integrations are more beneficial tools in expressing real-world matters than the integer-order ones. Various studies in the literature, on distinct fractional operators such as the classical Riemann-Liouville, Caputo, Katugamploa, Hadamard, and Marchaud versions have shown versatility in modeling and control applications across various disciplines. However, such forms of fractional derivatives may not be able to explain the dynamic performance accurately, hence, many authors are found to be sorting out new fractional differentiations and integrations which have a kernel depending upon a function and this makes the range of definition expanded [13,14]. Furthermore, models based on these fractional operators provide excellent results to be compared with the integer-order differentiations [15][16][17][18][19][20][21][22][23][24][25][26][27].
The derivatives in this calculus seemed complicated and lost some of the basic properties that usual derivatives have such as the product rule and the chain rule. However, the semigroup properties of these operators behave well in some cases. Recently, the authors in [28] defined a new well-behaved simple derivative called "conformable fractional derivative" which depends just on the basic limit definition of the derivative. It will define the derivative of higher-order (i.e., order δ > 1) and also define the integral of order 0 < δ ≤ 1 only. It will also prove the product rule and the mean value theorem and solve some (conformable) differential equations where the fractional exponential function e ϑ δ δ plays an important rule. Inequalities and their utilities assume a crucial job in the literature of pure and applied mathematics [29][30][31][32][33][34][35][36][37]. The assortment of distinct kinds of classical variants and their modifications were built up by using the classical fractional operators.
Convexity and its applications exist in almost every field of mathematics due to impermanence in several areas of science, technology in nonlinear programming and optimization theory. By utilizing the idea of convexity, numerous variants have been derived by researchers, for example, Hardy, Opial, Ostrowski, Jensen and the most distinguished one is the Hermite-Hadamard inequality [38][39][40][41].
Let I ⊂ R be an interval and Q : I → R be a convex function. Then the double inequality holds for all l 1 , l 2 ∈ I with l 1 l 2 . Clearly, if Q is concave on I, then one has the reverse of inequality (1.1). By taking into account fractional integral operators, several lower and upper bounds for the mean value of a convex function can be obtained by utilizing of inequality (1.1). Exponentially convex functions have emerged as a significant new class of convex functions, which have potential applications in technology, data science, and statistics. In [42], Bernstein introduced the concept of exponentially convex function in covariance formation, then the idea of an exponentially convex function is extended by inserting the condition of r-convexity [43]. Following this tendency, Jakšetić and Pečarić introduced various kinds of exponentially convex functions in [44] and have contemplated the applications in Euler-Radau expansions and Stolarsky means. Our aim is to utilize the exponential convexity property of the functions as well as the absolute values of their derivatives in order to establish estimates for conformable fractional integral introduced by Abdeljawed [45] and Jarad et al. [46].
Following the above propensity, we present a novel technique for establishing new generalizations of Hermite-Hadamard inequalities that correlate with exponentially tgs-convex functions and conformable fractional operator techniques in this paper. The main purpose is that our consequences, which are more consistent and efficient, are accelerated via the fractional calculus technique. In addition, our consequences also taking into account the estimates for Hermite-Hadamard inequalities for exponentially tgs-convex functions. We also investigate the applications of the two proposed conformable fractional operator to exponentially tgs-convex functions and fractional calculus. The proposed numerical experiments show that our results are superior to some related results.

Preliminaries
Before coming to the main results, we provide some significant definitions, theorems and properties of fractional calculus in order to establish a mathematically sound theory that will serve the purpose of the current article.
Awan et al. [47] proposed a new class of functions called exponentially convex functions.
Definition 2.1. (See [47]) A positive real-valued function Q : K ⊂ R → (0, ∞) is said to be exponentially convex on K if the inequality

1)
holds for all l 1 , l 2 ∈ R, α ∈ R and ϑ ∈ [0, 1]. Now, we introduce a novel concept of convex function which is known as the exponentially tgsconvex function.
The conformable fractional integral operator was introduced by Abdeljawad [45]. [45]) Let ρ ∈ (n, n + 1] and δ = ρ − n. Then the left and right-sided conformable fractional integrals of order ρ > 0 is defined by Next, we demonstrate the following fractional integral operator introduced by Jarad et al. [46].
Then the left and right-sided fractional conformable integral operators of order ρ > 0 are stated as: and Recalling some special functions which are known as beta and incomplete beta function.
Further, the following relationship holds between classical Beta and incomplete Beta functions:

Hermite-Hadamard type inequality for exponentially tgs-convex functions via conformable fractional integrals
Throughout the article, let I = [l 1 , l 2 ] be an interval in real line R. In this section, we shall demonstrate some integral versions of exponentially tgs-convex functions via conformable fractional integrals.
Some special cases of above theorem are stated as follows: Corollary 3.1. Choosing α = 0, then Theorem 3.1 reduces to a new result Remark 3.1. Choosing ρ = n + 1 and α = 0, then Theorem 3.1 reduces to Theorem 3.1 in [19].

Hermite-Hadamard type inequality for differentiable exponentially tgs-convex functions via conformable fractional integrals
Our next result is the following lemma which plays a dominating role in proving our coming results.
Lemma 4.1. For ρ ∈ (n, n + 1]) with ρ > 0 and let Q : I ⊂ R → R be differentiable function on I • (interior of I) with l 1 < l 2 such that Q ∈ L 1 ([l 1 , l 2 ]), then the following inequality holds: Proof. It suffices that Then by integration by parts, we have

Hermite-Hadamard inequality within the generalized conformble integral operator
This section is devoted to proving some new generalizations for exponentially tgs-convex functions within the generalized conformable integral operator.
Our main results depend on the following identity.
Lemma 5.2. For ρ > 0 and let Q : I ⊂ R → R be a differentiable function on (l 1 , l 2 ) with l 1 < l 2 such that Q ∈ L 1 [l 1 , l 2 ], then the following identity holds: Proof. It suffices that Using integration by parts and making change of variable technique, we have By substituting values of M 1 and M 2 in (5.8) and then conducting product on both sides by (l 2 −l 1 )ρ δ 2 , we get the desired result. Theorem 5.3. For ρ > 0 and let Q : I ⊂ R → R be a differentiable function on I • with l 1 < l 2 such that Q ∈ L 1 ([l 1 , l 2 ]). If | Q | r , with r ≥ 1, is an exponentially tgs-convex function, then the following inequality holds Proof. Using exponentially tgs-convexity of | Q | r , Lemma 5.2, and the well-known Hölder inequality, we have , the required result.
In this section, we use our obtained results in section 5 to provide several novel inequalities involving the special bivariate means mentioned above.
Proof. Let ρ = δ = 1 and Q(z) = z n . Then the desired result follows from Theorem 5.3.

Conclusions
In this paper, we proposed a novel technique with two different approaches for deriving several generalizations for an exponentially tgs-convex function that accelerates with a conformable integral operator. We have generalized the Hermite-Hadamard type inequalities for exponentially tgs-convex functions. By choosing different parametric values ρ and δ, we analyzed the convergence behavior of our proposed methods in form of corollaries. Another aspect is that to show the effectiveness of our novel generalizations, our results have potential applications in fractional integrodifferential and fractional Schrödinger equations. Numerical applications show that our findings are consistent and efficient. Finally, we remark that the framework of the conformable fractional integral operator, it is of interest to further our results to the framework of Riemann-Liouville, Hadamard and Katugampola fractional integral operators. Our ideas and the approach may lead to a lot of follow-up research.