On fractional biparameterized Newton-type inequalities

In this work, we present a novel biparameterized identity that yields a family of one-, two-, three-, and four-point Newton-type formulas. Subsequently, we establish some new Newton-type inequalities for functions whose first derivatives are α-convex. The investigation is concluded with numerical examples accompanied by graphical representations to substantiate the accuracy of the obtained results.


Introduction
Convexity is a fundamental notion in mathematics with broad-ranging applications in pure and applied fields.It plays a key role in the development of inequality theory, which has been an active area of research for many years.Integral inequality theory has found applications in many scientific fields, including physics, economics, and engineering.For some works related to Newton-type inequalities via different types of convexity, we refer the readers to [1][2][3][4][5][6].
Orlicz [7] introduced the class of α-convex functions as follows.
Definition 1.1 A function h : I ⊂ R + → R is said to be α-convex for some fixed 0 < α ≤ 1 if for all u, v ∈ I and t ∈ [0, 1], This notion, also called s-convexity in the first sense, represents a generalization of classical convexity, which can be recovered for α = 1.
Recently, fractional calculus has emerged as a powerful tool for modeling various physical and real-world phenomena.The idea of fractional derivatives and integrals has been around for centuries, but it is only in the last few decades that fractional calculus has received significant attention from mathematicians and scientists.The notions of fractional derivatives and integrals have been used in many areas of mathematics, including differential equations, complex analysis, and numerical analysis.For a comprehensive overview of the applications and recent developments in fractional calculus, we refer the readers to [8][9][10].
Many researchers have devoted their efforts to the study of integral inequalities via different fractional operators, and their works have led to significant advances in various areas of mathematics and beyond.In [11] the authors investigated Hermite-Hadamardtype inequalities using the Riemann-Liouville fractional integrals.Hamida et al. [12] established the fractional Bullen-type inequality.Sarikaya et al. [13] derived the fractional Simpson-type inequalities based on convexity, whereas Chen et al. [14] extended them to s-convexity.For further related works, we refer to [15][16][17][18][19][20][21] and the references therein.
More recently, Ali et al. [22] explored parameterized inequalities of Simpson 3/8 type for differentiable convex functions and established numerous classical and fractional inequalities of Newton type, trapezoidal type, and others.
In this paper, motivated by the above-mentioned results, we investigate a class of fractional inequalities, the biparametrized Newton-type inequalities for differentiable αconvex mappings.Our goal is to explore the general framework of quadrature formulas with one, two, three, and four points.To achieve this, we introduce a biparameterized identity involving the Riemann-Liouville fractional integrals, which enabled us to establish several new results for α-convex derivatives.We also review the existing literature on this topic and highlight the significance of our contributions.This study provides a plethora of new results, as well as some well-known ones, which can be applied in various fields of science and engineering.

Preliminaries
In this section, we review some well-known definitions and essential tools related to fractional calculus.

Definition 2.1 ([23]
) The beta function is defined for complex numbers u and v such that Re(u) > 0 and Re(v) > 0 by

Definition 2.2 ([23])
The incomplete beta function is defined for complex numbers u and v such that Re(u) > 0 and Re(v) > 0 and for 0 < c < 1 by

Definition 2.3 ([23]
) The hypergeometric function is defined for complex numbers u, v, w, and z such that Re(w) > Re(v) > 0 and |z| < 1 by where B(•, •) is the beta function.

Definition 2.4 ([23]
) Given h ∈ L 1 [υ, ], the Riemann-Liouville fractional integrals of order δ > 0 with υ ≥ 0, denoted by I δ υ + h and I δ -h, are defined as follows: respectively, where (δ) = ∞ 0 e -t t δ-1 dt is the gamma function, and The paper is organized as follows.In Sect.3, we introduce and prove a biparametric fractional identity, from which we establish a multitude of Newton-type inequalities with one, two, three, and four points for functions with α-convex derivatives.In Sect.4, we provide illustrative examples along with graphical representations to confirm the accuracy of the obtained results and showcase some applications.The study concludes with a final section summarizing our findings and providing a comprehensive conclusion.

Main results
Our approach begins by introducing the following identity, which plays a central role in establishing our results.
Theorem 3.2 presents a diverse array of findings regarding one-, two-, three-, and fourpoint Newton-type inequalities, encompassing both original discoveries and preexisting results.Notably, by scrutinizing select parameter values of x and λ, a wealth of new inequalities arises for various categories of convexity, and certain known ones are also reencountered, some of them even improved.We will enumerate a selection of such results.
Corollary 3.3 Setting α = 1 in Theorem 3.2, we obtain the following fractional biparameterized four-point Newton-type inequality for convex functions: Corollary 3.4 Setting x = υ in Corollary 3.3, we obtain the following fractional trapezoidaltype inequality for convex functions: Moreover, using the convexity of |h |, we obtain This improves the result obtained in Corollary 3.6 of [17].
Corollary 3.5 Setting x = υ+ 2 in Corollary 3.3, we obtain the following fractional parameterized three-point Newton-Cotes-type inequality for convex functions: Remark 3.6 Consider some particular cases in Corollary 3.5.
• If λ = 0, then we obtain the following fractional midpoint inequality for convex functions: This result improves that of Sarikaya et al. [11] (Theorem 5 for q = 1), which can be derived by using the convexity of |h | in the preceding inequality.• If λ = 1, then we obtain another variant of fractional trapezoidal-type inequality for convex functions, Moreover, using the convexity of |h |, we can recover the result established in Corollary 3.4.• If λ = 1  2 , then we obtain the following fractional Bullen-type inequality for convex functions: • If λ = 1 3 , then we obtain the following fractional Simpson-type inequality for convex functions: • If λ = 1 4 , then we obtain the following fractional Simpson-type inequality for convex functions: Corollary 3.7 Setting λ = 0 in Corollary 3.3, we obtain the following fractional symmetrical two-point type inequality for convex functions: Moreover, taking δ = 1, we obtain the following companion Ostrowski-type inequality for convex functions: Moreover, taking δ = 1, we obtain the following Simpson-type 3/8 inequality for convex functions: Corollary 3.11 In Corollary 3.10, taking α = 1, we obtain the following biparameterized Newton-type inequality for convex functions: Remark 3.12 In Corollary 3.11: • If we take x = υ, then we obtain Corollary 2 from [4].Moreover, if we use the convexity of |h |, then we obtain Theorem 2.2 from [3]. • If we take x = υ+ 2 and λ = 0, then we obtain Corollary 2 (for q = 1) from [6].Moreover, if we use the convexity of |h |, then we obtain Theorem 2.2 from [5].
In the following theorems, we will not provide specific cases, as these are left to the reader's curiosity.
Proof From modulus applied to Lemma 3.1, using the Hölder inequality, we have .
Using the α-convexity of |h | q , we get The proof is complete.
Here we have used ( 11)-( 14) and the fact that The proof is complete.

Illustrative examples
In this section, we presnt several two-and three-dimensional graphical representations to corroborate the accuracy of the results obtained in our study.Note that the graphical representations in this section were created using Matlab software, and in all the figures, the red color is used to depict the left-hand side (LHS), whereas blue is used for the right-hand side (RHS), consistently across all results.
Example 4.1 In this particular example, we initiate the analysis by establishing the parameter values of x = υ+ 2 = 1 2 and α = 1, and subsequently exhibit the outcomes as functions of the remaining parameters λ and δ.The function h defined over the interval [0, 1] as h(u) = u 2 is considered for this purpose.Accordingly, |h (u)| = 2u is a convex function over [0, 1], and hence Corollary 3.5 gives Figure 1 The comparative analysis between the left and right sides of Corollary 3.5

Figure 2 Particular cases
The surface plot of this result is depicted in Fig. 1.

Some particular cases:
• By fixing λ = 0 we obtain the curve related to the fractional midpoint rule with respect to δ, as shown in Fig. 2 (Case 1).• By fixing λ = 1 we obtain the curve related to the fractional trapezoidal rule with respect to δ, as shown in Fig. 2 (Case 2).• By fixing λ = 1  3 we obtain the curve related to the fractional Simpson rule with respect to δ, as shown in Fig. 2
The surface plot of this result is depicted in Fig. 3.

Particular case:
• By fixing λ = 3 8 we obtain the curve related to the Simpson-type 3/8 inequality via α-convexity, as shown in Fig. 4. The graphical representation of the results of these examples demonstrates the accuracy of the findings obtained in this study.

Conclusions
We have investigated a class of fractional biparameterized four-point Newton-type inequalities via α-convexity.By introducing a general identity involving Riemann-Liouville integral operators we have derived several new results that complement and extend the existing literature on this topic.Our contributions provide a broad range of tools that can be used in diverse fields of science and engineering.Furthermore, our findings can potentially have practical implications in numerical integration, optimization, and other related areas.
Note that from a purely practical standpoint, this result is used to choose the value of λ and possibly that of x to achieve the minimum of the right-hand side, which are obviously specific to the function h.
Finally, we hope that this work will inspire further research in this direction and lead to new applications and developments.

Figure 3 10 Figure 4
Figure 3The comparative analysis between the left and right sides of Corollary 3.10