A comprehensive study on Milne-type inequalities with tempered fractional integrals

In the framework of tempered fractional integrals, we obtain a fundamental identity for diﬀerentiable convex functions. By employing this identity, we derive several modiﬁcations of fractional Milne inequalities, providing novel extensions to the domain of tempered fractional integrals. The research comprehensively examines signiﬁcant functional classes, including convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation.


Introduction and preliminaries
Numerical integration is a crucial computational tool that tackles mathematical complexities without analytical solutions.It is used in various fields like physics, engineering, finance, economics, signal processing, computer graphics, probability, and statistics.Its versatility and efficiency make it essential in addressing various mathematical challenges across diverse fields [13,37].Researchers have contributed substantially to numerical integration by developing new formulas and studying error bounds in detail [16,17].In mathematical inequality, there is much research on finding new error bounds by using various types of functions, such as convex, bounded, Lipschitzian, and those with bounded variation [2,3].Numerous researchers have actively engaged with the realm of fractional calculus, particularly focusing on its implications in the theory of inequalities.In [8,31], the authors investigated fractional variants of trapezoid-type inequalities.Budak [6] studied midpoint and trapezoid-type inequalities for newly defined quantum integrals.For local fractional integrals, Sarikaya et al. [33] extended Gruss-and Chebysev-type inequalities on the fractal sets.Luo and Du [25] formulated equality and provided various results related to the Simpson-type inequality for Riemann-Liouville fractional integrals.By taking twice-differentiable functions, Hezenci et al. [20] gave a novel version of fractional Simpson-type inequalities.Recently, numerous publications have focused on the formation of significant inequities [1,4,21,38].
The Milne-type inequality, attributed to the British mathematician Edward Arthur Milne in the early twentieth century, is a significant mathematical tool for integral estimation.This inequality, widely acknowledged and bearing Milne's name, holds significance in mathematical inequalities, with applications extending to optimization theory, physics, and engineering [5,14,15,35].
Tempered fractional calculus, essentially evolved from the principles of fractional calculus, can be traced to the innovative study of Buschman [10], which introduced the concept of fractional integration involving weak singular and exponential kernels.Tempered fractional integrals have extensive applications in data processing, image advancement, bioengineering, finance, and other scientific fields [24,26].A significant leap in this field is attributed to Mohammed et al. [27], who notably formulated Hermite-Hadamard-type inequalities for convex functions using tempered fractional integrals.Their contributions extend beyond existing results in Riemann-Liouville fractional integrals, leveraging the methodology proposed by Sarikaya et al. [32] and Sarikaya and Yildirim [34].This technique provides a wide range of inequalities, including trapezoidal and midpoint inequalities, in the setting of tempered fractional integrals.Cao et al. [11] have established the equivalence of tempered fractional and substantial derivatives under certain conditions.Additionally, they obtained definitions and analyzed their properties.By employing tempered fractional integrals in the perspective of twice-differentiable functions, Hezenci and Budak [19] provided a novel identity.By involving this identity, they investigated various results related to left Hermite-Hadamard-type inequalities for tempered fractional integrals.
Numerous noteworthy fractional integrals, such as the Riemann-Liouville, Caputo, Grünwald-Letnikov, and Weyl fractional integrals, have been found in the literature.Numerous scholarly articles have been devoted to expanding and broadening the scope of these integrals.In particular, the relationships between Riemann-Liouville and tempered fractional integrals in [28], -Hilfer derivative [23], and (k)-fractional operators [29] were reported in the setting of tempered fractional integral.
The following section outlines the fundamental preliminaries required to prove our key results.
In the case of λ = 0, in Definition 3, immediately we acquire Definition 2. For an extensive review of tempered fractional integrals and their diverse cases, the following books offer extensive insights [26,30,36].
Budak et al. [9] investigated fractional forms of Milne-type inequalities for functions with bounded variation, Lipschitz, and differentiable convex functions.They were the first to investigate these inequalities, focusing on fractional integrals.
Leveraging previous investigations, we derive tempered fractional variations of Milnetype inequalities using differentiable convex mappings.We examine new bounds by involving differentiable convex mappings within the context of the tempered fractional integral.These resulting inequalities are versatile and can be transformed into Riemann-Liouville fractional Milne-type inequalities when λ = 0.Under these specified conditions, if we assume α = 1, the inequalities are reduced to basic Milne-type inequalities.

Main results
Initially, we establish an identity employing tempered fractional integrals.Subsequently, utilizing this particular identity, we derive new Milne-type inequalities with tempered fractional integrals.

Lemma 1 Consider
In this case, the following equality is valid: Proof Through the utilization of the integration by parts technique, we acquire By taking the same steps, we derive By ( 2) and (3), this yields The proof of Lemma 1 is concluded.
Theorem 1 Consider the conditions outlined in Lemma 1 and the convexity of the function |F | on the interval [ω, ς], then, we attain the following Milne-type inequalities for tempered fractional integrals: where Proof By applying the absolute value in Lemma 1 and considering the convexity of |F |, we acquire Consequently, the proof is concluded.
Theorem 2 Consider the conditions outlined in Lemma 1 and the convexity of the function |F | q , q > 1 on the interval [ω, ς].Then, we have the following Milne-type inequalities for tempered fractional integrals: 2 ) Here, q -1 + p -1 = 1 and Proof By utilizing Hölder's inequality in (6), this yields .
By utilizing the convexity of |F | q , we attain .
Theorem 3 Consider the conditions outlined in Lemma 1 and the convexity of the function |F | q , q ≥ 1 on the interval [ω, ς].Then, we have the following Milne-type inequalities for tempered fractional integrals: , where 1 (α, λ) is defined in Theorem 1, and Proof By employing the Power mean inequality in (6), we acquire .
Considering the convexity of |F | q on the interval [ω, ς], we derive . This concludes the proof.

Theorem 4 Consider the conditions outlined in Lemma
, then, we obtain the following Milne-type inequalities for tempered fractional integrals: where 1 (α, λ) is defined as in Theorem 1.
Proof From Lemma 1, it is easy to write By employing the properties of modulus in equation ( 10), we can derive Based on the given assumption m ≤ F (ξ ) ≤ M for ξ ∈ [ω, ς], this yields and With the utilization of inequalities ( 11) and ( 12), we achieve Hence, the proof is effectively concluded.
Theorem 5 Consider the conditions outlined in Lemma 1 to hold.If F is an L-Lipschitzian function on the interval [ω, ς], then we obtain the following Milne-type inequalities for tempered fractional integrals: L.
Proof Utilizing the modulus in Lemma 1, we attain Since |F | is an L-Lipschitzian function, we can conclude L.
Consequently, the demonstration is concluded.
Theorem 6 Assume F : [ω, ς] → R exhibits a bounded variation over the interval [ω, ς].Then, we have the following Milne-type inequality that is specifically tailored for tempered fractional integrals: Proof Define the mappings By employing the integration by parts technique, we acquire That is, It is a well-established fact that if g, F : [ω, ς] → R satisfy the conditions where g is continuous on [ω, ς] and F is of bounded variation on [ω, ς], then However, employing (15), we have 2 )  which is proposed in [3].
which is presented by Budak et al. in [9, Theorem 2].