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Some New Integral Inequalities Involving Fractional Operator with Applications to Probability Density Functions and Special Means

: Fractional calculus manages the investigation of supposed fractional derivatives and integrations over complex areas and their applications. Fractional calculus is the purported assignment of differentiations and integrations of arbitrary non-integer order. Lately, it has attracted the atten-tion of several mathematicians because of its real-life applications. More importantly, it has turned into a valuable tool for handling elements from perplexing frameworks within different parts of the pure and applied sciences. Integral inequalities, in association with convexity, have a strong relationship with symmetry. The objective of this article is to introduce the notion of operator reﬁned exponential type convexity. Fractional versions of the Hermite–Hadamard type inequality employing generalized R − L fractional operators are established. Additionally, some novel reﬁnements of Hermite–Hadamard type inequalities are also discussed using some established identities. Finally, we present some applications of the probability density function and special means of real numbers.


Introduction
In the recent past, the theory of inequalities via different types of convexities has made significant contributions to many areas of mathematics. The theory of inequalities plays a significant role in various branches of mathematics such as optimization, differential equations, functional analysis, probability, numerical analysis, finite element method, fractional calculus, etc. Moreover, convexity implies a significant interest in mathematical inequalities because of the nature of its definition. Variants of convex functions play a critical role in several branches of sciences such as biological systems, economics, and optimization. Mathematical inequalities for various convex functions have been generalized colossally, having a critical impact on traditional investigation. Symmetry, convexity, and fractional operator have a very strong connection because of their fascinating properties. Whichever one we work on, it can be applied to the accompanying one because of the solid relationship passed on between them. One can refer to the references [1][2][3][4][5][6][7][8] for different types of convexities and related inequalities.
Tunç et al. [12] established new versions of Hermite-Hadamard inequality using both classical as well as fractional integral operators for tgs-convex functions as follows: Let G : I −→ R be a tgs-convex function on I in R and d 1 , d 2 ∈ I with d 1 < d 2 , then, The coressponding Hermite-Hadamard inequality via R-L fractional operator is expressed as: Let G : I ⊆ R be a positive function with d 1 < d 2 and L 1 [d 1 , d 2 ]. If G is a tgs-convex function on [d 1 , d 2 ], then the following inequalities for fractional integrals hold: .
For some recent generalizations on various forms of convex functions, we refer interested readers to see [13][14][15][16] and the references cited therein.
After reviewing the hypothesis of convexity and its generalizations in the theory of inequality, we learned about a new class of convexity called exponential convexity. Due to its applications in various fields such as big-data analysis, deep learning, and information theory, several researchers have shown their deep interest in exponential convexity. Study on big-data analysis and deep learning has recently expanded the adequacy of information theory involving exponentially convex functions (see [17,18]). Especially in the last few decades, different mathematicians worked on the idea of exponential-type convexity in various directions and contributed to the field of analysis such as Jakšetić et al. [19], Dragomir et al. [20], and Awan et al. [21] who presented Hermite-Hadamard type inequalities for exponential convex functions. Saima et al. [22] proved fractional versions of the Hermite-Hadamard inequality for exponential convex functions. Furthermore, Noor et al. [23] generalized the exponential convex function to the exponential preinvex function and discussed some of its properties. In 2020, Kadakal et al. [24] introduced a new notion of exponential type convex function and generalized some known integral inequalities.
Kadakal et al. [24], introduced the concept of exponential type convex function that generalizes convex functions.

Definition 2.
[24] A function G : I −→ R ⊆ R is said to be an exponential type convex function, if G is non-negative, for all d 1 , d 2 ∈ I and u ∈ [0, 1] , we have Very recently Jung et al. [25], introduced the concept of refined (α, h − m) convex function, given as The above definition generalizes some well-known convexities such as (α, m)-convex function, m-convex function, (s, m)-convex function, refined h-convex, refined (h − m) convex, refined (s, m)-convex functions, etc.
In the last decade, fractional calculus has gotten a lot of consideration. This topic has attracted the interest of many researchers because of its broad applications in different fields such as probability theory, biomathematics, image processing, fluid mechanics, material science, viscoelasticity and designing, etc. In recent times it is seen that several mathematicians utilize their notations and approaches to study a variety of definitions that fit the possibility of fractional-order integrals and derivatives. The form that is discussed most in the realm of fractional calculus is the R-L operator and its variants. It is interesting to note that the R-L meaning of a fractional derivative gives us the same outcome as that acquired by Lacroix [26].
In the modern era, fractional analysis and inequality theory have developed together. A fundamental component of applied sciences and mathematics is fractional calculus. Academics urge many students to think about applying fractional calculus to solve difficulties in the real world. The Hermite-Hadamard type integral inequalities [27], Hermite-Hadamard-Mercer inequalities [28], the Ostrowski inequality [29], and the Simpson type inequality [30] have all been studied using the Riemann-Liouville fractional integral operators. The Simpson-Mercer integral inequality was studied utilizing the Atangana-Baleanu fractional operator in [31]. The Hermite-Hadamard inequality and the Fejér type integral inequalities were investigated via Katugampola type fractional integral operators in [32]. The Hermite-Hadamard inequality and its Mercer equivalent were also examined using the Caputo-Fabrizio fractional integrals [33,34]. The data described above demonstrates the strong connection between integral inequalities and fractional operators.
To additionally encourage the conversation started in this article, we present the definition of the R-L fractional operator and ψ-R-L fractional operator.
The following identities will be used in the results to follow.
The fractional integral inequalities, the definition of refined (α, h − m) convex, and exponential type convexity are the motivation of our results for this paper. In this article, a new notion of a generalized exponential convex function, i.e., refined exponential type convex function is studied. Employing this new notion, we generalized the H − H inequality via both classical integral and fractional integral operators. Some novel refinements of the H − H type inequality via generalized R − L and ψ − R − L fractional integral operators are discussed as well. This article also deals with applications to the probability density function and special means.

Refined Exponential Type Convexity and H − H Type Inequalities
Now, we introduce a new notion of convex function, i.e., a refined exponential type convex function, and establish some results based on the said convexity.

Definition 5.
A function G : I −→ R is said to be a refined exponential type convex function if for every d 1 , d 2 ∈ I and u ∈ (0, 1) hold true for all u ∈ (0, 1).

Proposition 1.
Every non-negative tgs convex function is a refined exponential type convex function.
Proof. Using Lemma 1, we have u ≤ e u − 1 and 1 − u ≤ e 1−u − 1. This implies that Next, we prove the classical version of the Hermite-Hadamard inequality corresponding to the introduced new notion of refined exponential type convex function.
. If G is a refined exponential type convex function, then the following inequality holds: As G is a refined exponential type convex function, we have Consequently, putting u = 1 This completes the proof of the first part and to complete the second part, we use the definition of refined exponential type convex function: Using the change of the variable technique and then integrating with respect to u over [0,1], we obtain From (8) and (9) , we establish the desired result (6). Now, we prove some fractional versions of the Hermite-Hadamard inequality corresponding to the refined exponential type convex function via Riemann-Liouville and ψ-Riemann-Liouville fractional integral operator. The proven results show that we can apply different fractional operators for these types of convexities. The results also show the application of hypergeometric functions for inequalities as well.

Fractional Inequalities of H − H Type
. If G is a refined exponential type convex function, then the following inequality holds: Proof. As G is a refined exponential type convex function, one has (11), then multiplying both sides of the resultant inequality by u λ−1 , and finally integrating w.r.t to u over [0, 1], we obtain This proves the first part of the theorem. For the second part of the inequality we use the definition of refined exponential type convexity of G, i.e., Adding the last two inequalities and then following the same procedure, we obtain Using definition 3, we have This completes the proof of Theorem 5.
It is evident from articles [34,36,[43][44][45][46]] that many researchers are now focusing on the mid-point type inequalities corresponding to the Hermite-Hadamard inequality. Our next result is aimed toward this as well.
. If G is a refined exponential type convex function, then the following inequality holds: Proof. Since G is a refined exponential type convex function, one has Putting (15) and then multiplying both sides by u λ−1 and finally integrating with respect to u over [0, 1], we have Consequently, This completes the proof of the first part. Next, to prove the second part we use the definition of refined exponential type convexity of G, i.e., Upon adding the last two inequalities and then following the same procedure as above, we obtain Equations (16) and (17) lead to the proof of Theorem 5.

Further Estimates on H − H Inequalities
In the following theorems, we prove some trapezoidal type inequalities with the help of some classical inequalities such as Hölder's inequality, Young's inequality, and power mean inequality via Riemann-Liouville fractional operator.
Then the following equality holds: Then the following equality holds: − G be R-L fractional operators. If |G | is a refined exponential type convex function, then the following inequality holds: Proof. Using Lemma 2, the refined exponential type convexity of |G|, we have This completes the proof.
is a refined exponential type convex function, then the following inequality holds: Proof. Using Lemma 2, the refined exponential type convexity of |G|, we have This completes the proof. − G be R-L fractional operators. For q ≥ 1 , if |G | q is a refined exponential type convex function, then the following inequality holds: Proof. Using Lemma 2, the refined exponential type convexity and power-mean inequality, we have This completes the proof.
and is a refined exponential type convex function, then the following inequality holds: Proof. Taking Lemma 3 and the refined exponential type convexity of |G| into consideration, we have and |G| q is a refined exponential type convex function, then the following inequality holds: Proof. Taking Lemma 3, Hölder's inequality, and the refined exponential type convexity of |G| q into consideration, we have

Inequalities via Generalized R-L Fractional Integral Operator
Here, we intend to establish new results for a generalized fractional integral operator, i.e., ψ-Riemann-Liouville fractional integral operator and inequalities such as Hölder's inequality and power mean inequality to show the efficiency of the main results.
, then the following inequality for fractional integral holds Proof. Since G is a refined exponential type convex function, one has Putting x = ud 1 + (1 − u)d 2 and y = ud 2 + (1 − u)d 1 in (20), multiplying both sides by u λ−1 , and then integrating with respect to u over [0, 1], we obtain Consider, The proof of the first part is completed, next to prove the second part we use the definition of refined exponential type convexity: Upon adding the last two inequalities and following the same procedure as above, we obtain Again, from the proof of the first inequality This completes the proof of Theorem 11.
, then the following inequality for fractional integral holds Proof. Since G is an exponential type convex function, one has Putting x = ud 1 + (1 − u)d 2 and y = ud 2 + (1 − u)d 1 in (23), multiplying the resultant inequality by u λ−1 , and then integrating with respect to u over [0, 1], we obtain Consider, This proves the first part of the theorem; next to prove the second part, we use the definition of refined exponential type convexity, i.e., Upon adding the last two inequalities and then following the same procedure as above, we have Further computations give Multiplying bt λ 2 , we obtain the desired result.
, a positive monotonically increasing function on (d 1 , d 2 ] with ψ (v) being continuously differentiable on (d 1 , d 2 ) and λ ∈ (0, 1). Then, the following equality holds: be ψ-R-L fractional operators and ψ(v) be a positive increasing function on (d 1 , d 2 ] with ψ (v) being continuously differentiable on (d 1 , d 2 ) and λ ∈ (0, 1). If G : [d 1 , d 2 ] −→ R is a differentiable mapping on (d 1 , d 2 ) and |G | q is a refined exponential type convex function, then the following inequality holds: Proof. Using Lemma 4 and the refined exponential type convexity, Now, using Hölder's inequality This completes the proof. d 2 ), q ≥ 1, and |G | q is a refined exponential type convex function, then the following inequality holds: Proof. Using Lemma 4 and the refined exponential type convexity, Now, using the power-mean inequality This completes the proof.
, q ≥ 1, and |G | q is a refined exponential type convex function, then the following inequality holds: Proof. Using Lemma 4 and the refined exponential type convexity, Now, using the power-mean inequality Now, using Hölder's inequality This completes the proof.

Proposition 2.
Taking the assumptions given in Theorem 3 into consideration, we have Proof. Let G = Ω in Theorem 3 and considering the following into account Ω(u)du.

Application to Special Means
In this section we apply our results to establish several new inequalities for special means. Arithmetic mean : Generalized logarithmic mean : Proposition 3. Let d 1 , d 2 ∈ (0, ∞) with d 1 < d 2 . Then, Proof. Proposition 3 follows directly from Theorem 12 for the function G(x) = x 2 , ψ(x) = x and α = 1.
Let G(x) = x n , ψ(x) = x, and α = 1, then we have a general result.

Conclusions
In recent years, the use of fractional calculus to prove various integral inequalities using different convex functions has surged. Recent developments in the areas of differential equations, modeling, and mathematical inequalities all benefit from fractional operators. In this article, we focused on introducing the notion of refined exponential type convexity and presenting the fractional Hermite-Hadamard inequality and its refinements. The ability to demonstrate inequalities of the H − H type on coordinates, quantum calculus, and interval-valued analysis using the ideas discussed in this article will be an intriguing test of their viability.

Data Availability Statement: Not applicable
Acknowledgments: The authors would like to thank Khon Kaen University, Thailand.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: