New error bounds for Newton’s formula associated with tempered fractional integrals

In this paper, we ﬁrst construct an integral identity associated with tempered fractional operators. By using this identity, we have found the error bounds for Simpson’s second formula, namely Newton–Cotes quadrature formula for diﬀerentiable convex functions in the framework of tempered fractional integrals and classical calculus. Furthermore, it is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature


Introduction
One of the most famous inequalities for convex functions is Hermite-Hadamard-type inequality on account of its rich geometrical importance and applications.Thus, remarkable number of mathematicians have considered the Hermite-Hadamard-type inequalities and related these inequalities such as trapezoid, midpoint, and Simpson's inequality.In last decades, the fractional calculus has application areas in various fields such as engineering, chemistry, and physics as well as mathematics.Because of its basic properties and applications in domains of science, fractional calculus has been the center of attraction in applied and pure mathematics.The application of arithmetic carried out in classical analysis in fractional analysis is very significant in order to obtain more realistic results in the solution of many problems.It can be established the bounds of new formulas by using not only Hermite-Hadamard and Simpson type inequalities but also Newton-type inequalities.Because of the importance of fractional calculus mentioned in this paragraph, one can examined distinct fractional integral inequalities extensively.While integer orders are a model that is not suitable for nature in classical analysis, fractional computation in which arbitrary orders are examined enables us to obtain more realistic approaches.
Simpson's second rule has the rule of three-point Newton-Cotes quadrature, thus evaluations for the case of three steps quadratic kernel are generally called Newton-type results.These results are also known as Newton-type inequalities in the literature.Many researchers have been investigated to Newton-type inequalities extensively.For example, in paper [1], some Newton-type inequalities for the case of functions whose first derivative in absolute value at certain power are arithmetically-harmonically convex.In addition to this, some new Newton-type inequalities for the case of differentiable convex functions involving Riemann-Liouville fractional integrals were proved in paper [2].Moreover, the authors also presented some Newton-type inequalities including Riemann-Liouville fractional integrals for functions of bounded variation.Furthermore, new Newton-type inequalities based on convexity were given in paper [3].It can be referred to [4][5][6][7][8][9][10][11], and the references therein to the case of more informations.
Tempered fractional calculus can be specified as the generalization of fractional calculus.The definitions of fractional integration with weak singular and exponential kernels were firstly reported in Buschman's earlier work [12].See the books [13][14][15] and references therein for more information about the different definitions of the tempered fractional integration.Mohammed et al. [16] established several Hermite-Hadamard-type inequalities connected with the tempered fractional integrals for the case of convex functions, which cover the previously published result such as, Riemann-Liouville fractional integrals.
This paper is organized with respect to the following plans: In Sect.2, the fundamental definitions of fractional calculus and other relevant research in this discipline are presented.In Sect.3, we prove an integral equality that is critical in establishing our primary results.With the help of this identity, we establish several Newton-type inequalities involving the tempered fractional integrals.In Sect.4, we provide our results by using special cases of obtained theorems.In other words, we find the error bound of Newton's rule with the help of the obtained results.Finally, in Sect.5, summary and concluding remarks are noted.
We shall now give the basic definitions and new notations of tempered fractional operators.

Theorem 3 Let us consider that the assumptions of Lemma 1 are valid and the function
. Then, the following Newton's rule inequality holds: where Proof We shall first take modulus in Lemma 1.Then, we get λ(σ 2 -σ 1 ) (α, 1) With the aid of the convexity of |F |, it follows This completes the proof of Theorem 3.
Theorem 4 Suppose that the assumptions of Lemma 1 hold.Suppose also that the function . Then, we have the following Newton's rule inequality .
Remark 3 If we choose α = 1 and λ = 0 in Theorem 3, then the following Newton-type inequality holds: which is given by [9,Remark 3].This inequality helps us to find the error bound of Newton's rule.
Remark 5 If we assign α = 1 and λ = 0 in Theorem 4, then we have the following Newtontype inequality , which is given by Sitthiwirattham et al. in [9,Remark 5].

Summary & concluding remarks
In this paper, we first establish an integral equality connected with tempered fractional operators.With the help of this equality, we have found the error bounds for Simpson's second formula, namely Newton-Cotes quadrature formula for differentiable convex functions in the framework of tempered fractional integrals and classical calculus.More precisely, with the help of the Hölder and power-mean inequality, we prove several Newtontype inequalities involving tempered fractional operators.Furthermore, some results are presented by using special choices of obtained inequalities.These type of inequalities will inspire new studies in various fields of mathematics.In the future works, mathematicians can try to generalize our results by utilizing a different version of convex function classes or another type fractional integral operators.Moreover,