Some novel refinements of Hermite-Hadamard and Pachpatte type integral inequalities involving a generalized preinvex function pertaining to Caputo-Fabrizio fractional integral operator

: In this article, we aim to introduce and explore a new class of preinvex functions called n - polynomial m -preinvex functions, while also presenting algebraic properties to enhance their numerical significance. We investigate novel variations of Pachpatte and Hermite-Hadamard integral inequalities pertaining to the concept of preinvex functions within the framework of the Caputo-Fabrizio fractional integral operator. By utilizing this direction, we establish a novel fractional integral identity that relates to preinvex functions for di ff erentiable mappings of first-order. Furthermore, we derive some novel refinements for Hermite-Hadamard type inequalities for functions whose first-order derivatives are polynomial preinvex in the Caputo-Fabrizio fractional sense. To demonstrate the practical utility of our findings, we present several inequalities using specific real number means. Overall, our investigation sheds light on convex analysis within the context of fractional calculus.

attention due to its practical applications and has attracted mathematicians' interest. Sarikaya et al. [23] introduced a fractional Hermite-Hadamard inequality. Fractional operators play a crucial role in the development of fractional calculus, and the study of fractional integral inequality has been motivated by the exploration of well-known inequalities like Ostrowski, Simpson, and Hadamard. Fractional calculus finds application in various fields, such as transform theory, engineering, modeling, finance, mathematical biology, fluid flow, natural phenomena prediction, healthcare, and image processing. The references [24][25][26][27] provide further insights into this topic.
In [28] a comprehensive and up-to-date review on Hermite-Hadamard-type inequalities for different kinds of convexities and different kinds of fractional integral operators is presented. In this manuscript, we aim to examine the fractional Hermite-Hadamard and fractional Pachpatte-type inequalities related to polynomial preinvexity pertaining to Caputo-Fabrizio fractional integral operator. The Caputo-Fabrizio fractional operator introduced by Caputo and Fabrizio in [29], features a nonsingular kernel in its fractional derivatives and does not rely on the Gamma function. Notably, this operator allows any real power to be transformed into an integer order using Laplace transformation, enabling solutions to various related problems. This operator has found applications in modeling COVID-19 [30,31], Hepatitis B epidemic model [32], financial [33], ground water flow [34], and more.
In light of the preceding topic and as motivated by the significant research initiatives, the conceptualization of this work is organized in the following ways. First of all, in Section 2, we add some recognized definitions, theorems and remarks because all these are required in upcoming subsequent sections. In Section 3, we introduce the new definition namely n-polynomial m-preinvex function. Further, we add in this section its algebraic properties. In Section 4, we prove the Hermite-Hadamard inequalities via newly introduced concept via Caputo-Fabrizio fractional operator. In Section 5, we examine a new lemma and utilizing this newly introduced lemma with the help of Hölder and power mean inequality, we show some new variants of Hermite-Hadamard type inequality via newly introduced definition utilizing Caputo-Fabrizio fractional operator. In Section 6, employing the above equality with the help of improved version Hölder and power mean inequality, we attain some new variants of Hermite-Hadamard type inequality via newly introduced definition utilizing Caputo-Fabrizio fractional operator. In Section 7, we examine a new sort of Pachpatte type inequality via Caputo-Fabrizio fractional operator via newly introduced definition. Next, in Section 8, we offer some applications in the manner of constructed results. Finally, in Section 9, we give a conclusion.

Preliminaries
In this section, it would be appropriate to examine and concentrate on a few theorems, remarks and definitions, for the benefit of the readers' attention, clarity, relevance, and quality. This section's main objective is to research and discuss specific related ideas and concepts that are pertinent to our analysis in later sections of this paper. In the context of classical calculus, the subject convexity and the generalized form of the H-H inequality are initially studied and investigated. Additionally, we reviewed the m-preinvex function, Sobolev space, extended condition C and C-FFIO. We finish this part by reviewing the importance of the G − m PF for our research.
Jensen first time introduced the term convexity in the following manner: Theorem 2.1. (see [37]) Let D : I ⊆ R → R be a convex function with d 1 < d 2 and d 1 , d 2 ∈ I. Then the following inequality holds true: For some recent estimations, see [38][39][40]. , 0) (0, 1 2 ] and The above example is valid only for m-invex set not for convex set.
Recently, Deng [42] introduced generalized m-preinvex function, which is defined as: The following generalized Condition C first time introduced by Ting Song Du [43] in the aspect of m-preinvex. Extended Condition-C: We provide a few essential definitions from fractional calculus theory that will be utilised in the results that follow.
In this manuscript, we extend the equality utilizing C-FFIO, which is introduced by Dragomir [51].
Definition 2.6. ( [52]) Let D be a non-negative and real-valued function, Then D is n-polynomial preinvex, if

Generalized preinvex functions and its algebraic properties
Convexity has emerged as a captivating and practical field of study in both applied and pure sciences. The investigation of convex functions has led to innovative methods and calculations, providing a meaningful structure for addressing complex mathematical problems. The relationship between convexity and inequalities has garnered significant interest, leading to the exploration of various versions of classical inequalities.
Here, we are going to address the G − m PF, an intriguing idea for preinvex functions, and look at some of its algebraic features and examples.
(iii) Choosing n = m = 1 and N(d 1 , d 2 , m) = d 1 − md 2 , then Definition 3.1 collapses to the concept of convex function which was discussed by Niculescu [54]. Proof. The proof is obvious. Proof. Employing to the definition of m-preinvexity and Lemma 3.1, we have .
Proof. Utilizing the property of G − m PF and given condition, we have From the preceding proposition, it seems obvious that the newly developed preinvexity is very large in comparison to already published functions, such as convex and preinvex. This is the most appealing feature of the intended new Definition 3.1.
Next, we show some examples in manner of the newly introduce idea.
Example 3.1. D(d) = |d| ∀d ≥ 0 is a convex function, implies that the given function is a preinvex function (see [55]). Further, implies that, it is m-preinvex function if m = 1. By utilizing Proposition 3.1, it is an G − m PF .
Example 3.2. D(d) = e d ∀d ≥ 0 is a convex function, implies that the given function is preinvex function (see [55]). Further, implies that, it is m-preinvex function if m = 1. By utilizing Proposition 3.1, it is an G − m PF . Now we will investigate and expound on several examples of the newly introduced notion. We can see that D(x) = −|x| is preinvex but not convex.  Proof. Since given that D and D are two G − m PF, then This is the required proof.
Proof. Since given that D is G − m PF and c is any constant number, then This completes the proof. .
Proof. Given that D is an G − m PF w.r.t N for m ∈ (0, 1] and s ∈ [0, 1], and assuming that D(d 1 ) ≤ D(d 2 ), for all d 1 , d 2 ∈ X, we have In the same manner, if we let D(d 2 ) ≤ D(d 1 ) for all d 1 , d 2 ∈ X, we can also get Consequently, That is, D : X ⊆ R n → R is generalized quasi m-preinvex function on m-invex set X with respect to N. is also an G − m PF on A w.r.t N for m ∈ (0, 1], where d u is constant.
Proof. The proof is obvious.
and s ∈ [0, 1]. Assume that D is monotonic decreasing, N is monotonic increasing regarding m for Combining the monotone decreasing of the function D with the monotone increasing of the mapping N regarding m for fixed d 1 , d 2 ∈ R 0 and m 1 ≤ m 2 , it follows that This completes the proof.

H-H inequality involving n-polynomial m-preinvex function via Caputo-Fabrizio operator
Since the notion of convex function was first introduced more than a century ago, an enormous number of outstanding inequalities have been proven in the domian of the convex theory. The most widely recognized and frequently utilized inequality in the field of convex theory is the H-H inequality. Hermite and Hadamard were the ones who first suggested this inequality. Many mathematicians were motivated by the idea of this inequality to investigate and analyze the classical inequalities utilizing the many convexity senses. For instance, Xi [56], Mehreen [57], and Kirmaci [58] presented several kind of this inequality via convex functions. Hudzik [59], Dragomir [60] and Ozcan [61] addressed the notion of s-convex function and proved a novel variant of this inequality. Butt [62] and Rashid [63] proved this inequality in the polynomial version involving a new class of convexity.
The principal goal of this section is to use the G − m PF and the C-FFIO to derive and demonstrate the H-H inequality.
and satisfies extended condition-C. Then Proof. Employing the property of G − m PF of D, we have that This completes the right side inequality. For left side, utilizing the definition of G − m PF and extended condition C for N and integrating over [0, 1] we get This completes the proof.
, then the following H-H type inequalities hold: Proof. Employing the definition of G − m PF , it follows from the Inequality (4.1) that Multiplying both sides of (4.2) by ωN(d 2 ,d 1 ,m) On the other hand, from the Inequality (4.1), we have and add 2(1−ω) B(ω) D(k) to the resulting inequality, we obtain Combining the above Inequalities (4.4) and (4.6), we obtain the desired result.

Refinements of H-H type inequalities involving power mean and Hölder inequality via Caputo-Fabrizo operator
In the topic of convex analysis, a number of researchers have lately contributed on unique approaches to this challenge from various viewpoints. Recent research on H-H inequalities for convex functions has led to a wide range of innovations and improvements. Noor [65] utilized the notion of preinvex function and demostrated a new sort of H-H inequality. After Noor's published idea, many mathematicains set up new estimations of this inequality in the aspect of numerous kind of preinvexity. For example, Barani et al. [66] first time derived some refinements and estimations of this inequality for functions whose derivative of absolute values are preinvex. Noor [67] proved this inequality and its refinements pertaining to h-preinvexity. For the identical and similar preinvexity notions, we discuss to Wu et al. [68], Park [69], Sarikaya et al. [70], and Wang and Liu [71].
First, in this section, we look into the preinvex function lemma. We will include the results with the help of Hölder and power-mean inequality based on the recently investigated lemma. We add a few corollaries and observations in this section to increase the importance and caliber of this section.

Multiplying both sides with
This completes the proof.
Proof. Employing the Lemma 5.1, we have Employing the property of G − m PF, we have This completes the proof.
Corollary 5.1. Choosing n = 1, then we have Corollary 5.2. Assume that m = 1. Then we have Corollary 5.3. Choosing n = m = 1, then we have Corollary 5.5. If we put m = 1 and N( Corollary 5.6. Assume that n = 1 and N( Remark 5.1. If we put N(d 2 , d 1 , m) = d 2 −md 1 and n = m = 1, then we get the Theorem 5 in published article [45].
Proof. Employing Lemma 5.1, we have Employing Hölder inequality, we have This is the required proof.
Employing power mean inequality, we have Employing the property of G − m PF of |D ′ | q , we have By simplifying we obtain the desired result.

Refinements of Hermite-Hadamard type inequalities using Hölder Iscan and improved power mean inequality via Caputo-Fabrizo operator
In the area of integral inequalities, several mathematicians have lately collaborated on novel strategies to this subject from numerous viewpoints and multiple views. In 2019, the Hölder Iscan integral inequality was studied byİşcan [72] for the first time as estimations of Hölder inequality. The refined version of power mean inequality is called the improved power mean integral inequality. This inequality was first time explored by Kadakal [73] in 2019.
The major goal of this aspect is to achieve fresh findings using Hölder-Iscan inequality and improved power mean inequality. Here, we add some remarks and corollaries for value and worth.
Employing Hölder-İşcan inequality, we have This is the required proof.
Proof. First, assume that q > 1. According to the Lemma 5.1, we have Utilizing the statement of the improved power mean inequality, we have This is the required proof.
Corollary 6.8. Choosing n = 1 in the Inequality (5.2), we attain Corollary 6.9. Choosing m = 1 in the Inequality (5.2), we attain Corollary 6.10. Choosing n = m = 1 in the Inequality (5.2), we attain Corollary 6.11. If we put N(d 2 , d 1 , m) = d 2 − md 1 in the Inequality (5.2), then Corollary 6.12. If we put N(d 2 , d 1 , m) = d 2 − md 1 and m = 1 in the Inequality (5.2), then Corollary 6.13. Assume that n = 1 and N(d 2 , d 1 , m) = d 2 − md 1 in above Theorem. Then Corollary 6.14. Assume that n = m = 1 and N(d 2 , d 1 , m) = d 2 − md 1 in above Theorem. Then 7. Pachpatte type (h, m)-preinvex via n-polynomial m-preinvex function pertaining to Caputo-Fabrizio fractional integral operator The term "convexity" has drawn substantial interest in the past couple of decades due to its relevance and recognition of the concept of inequality. Due to developments of the convexity in applied sciences, many inequalities have been proposed and identified in the realm of convex analysis. Preinvexity has also been discussed by a large number of mathematicians, and numerous books have been written that provide new estimates and generalizations. With the help of these studies and research, the amazing Pachpatte-type inequality in the aspect of preinvexity is greatly enhanced. Preinvexity is an essential notion in the formation of extended convex programming. Nian Li [74] utilized the concept of time scales in 2009 and attained the novel variants of Pachpatte-type integral inequalities. In 2021, Butt [75] first time addressed fractional Pachpatte-Mercer-type inequalities in the sense of harmonic convexity. In 2022, Sahoo [76] utilized the idea of interval analysis and attain the novel sorts of Pachpatte-type integral inequalities in the aspect of center-radius order via preinvexity. This inequality in the mode of fractional operator associated with concept of exponential kernel is addressed by Sahoo [77] in 2022. Tariq et. al. [78] employed the non-conformable operator and attained a new kind of Pachpatte-type inequality via generalized preinvexity.
In light of the foregoing literature, we will investigate and research the Pachpatte-type inequality for the C-FFIO. A corollary and many remarks are added to heighten the relevance and worth of this section.

Applications to means
The subject convex analysis and fractional mathematics are both utilized in applied sciences. The literature makes it clear that these ideas have a broad spectrum of potential uses in multiple fields of research, from fluid dynamics to optimization. In order to be more precise, we are going to apply certain mean-type inequalities, such as arithmetic, geometric, and harmonic means inequalities, to the H-H inequality associated with the C-FFIO via G − m PF. The following mean-type inequalities have remarkable utilization in the domains of probabilities, statistics, circuit theory, stochastic processes, engineering, numerical approximations, and machine learning. In this section, we investigate the means as applications for two positive number d 1 , d 2 with d 1 < d 2 , which are given as: (1) The arithmetic mean (2) The generalized logarithmic mean Now employing the results in part 5, we investigate several inequalities involving special means. So here, we take B(ω) = B(1) = 1.

Conclusions
Fractional calculus has a greater influence and provides more precise results when examining computer models. Fractional calculus is widely utilized in applied mathematics, mathematical biology, engineering, simulation, and inequality theory. Numerous researchers across multiple scientific domains have expressed a keen interest in fractional calculus. In this paper: 1) First, we explored a novel type of preinvex function namely G − m PF. Further, we added some algebraic properties regarding this newly introduced definition. 2) We proved some refinements of the H-H and Pachpatte type inequalities via G − m PF in the aspect of the C-FFIO. 3) In addition, a novel integral identity is presented and several results in the sense of C-FFIO are attained via a newly introduced concept. 4) To improve the reader's interest and overall quality, we showed the refinements of H-H inequality regarding the newly introduced lemma with the help of improved power mean and Hölder Iscan inequality. 5) Some corollaries and remarks are added. 6) Finally, some meaningful applications regarding newly introduced ideas are explored. This paper's new notion can be extended to numerous inequalities employing the fractional H-H. Further generalizations can be made using some fresh ideas, such as interval-valued R-L convexities, center-radius order convexities and fuzzy interval convexities. To observe the behavior of this inequality, excited and inspired mathematicians can also use interval-valued functions and quantum calculus, etc. The investigation of fractional versions of inequalities using various new convex function types will be very intriguing to watch.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.