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

Abstract

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.

1 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], Ψ-Hilfer derivative [23], and $(\Psi -k)$-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.

Definition 1

(See [27])

For the real numbers $\alpha >0$ and $x, \lambda \geq 0$, the λ-incomplete gamma function is described as:

$$ \curlyvee _{\lambda}(\alpha , x):= \int _{0}^{x} \xi ^{\alpha -1} e^{- \lambda \xi} \,d\xi . $$

If $\lambda =1$, it is equivalent to the incomplete gamma function [12]:

$$ \curlyvee (\alpha , x):= \int _{0}^{x} \xi ^{\alpha -1} e^{-\xi} \,d\xi . $$

Here, $0<\alpha <\infty$ and $\lambda \geq 0$.

Remark 1

(See [27])

For the real numbers $\alpha >0$ and $x, \lambda \geq 0$, we attain

• I. $\curlyvee _{\lambda (\varsigma -\omega )}(\alpha , 1)=\int _{0}^{1} \xi ^{\alpha -1} e^{-\lambda (\varsigma -\omega ) \xi} \,d \xi = \frac{1}{(\varsigma -\omega )^{\alpha}} \curlyvee _{\lambda}(\alpha , \varsigma -\omega )$.

• II. $\int _{0}^{1} \curlyvee _{\lambda (\varsigma -\omega )}(\alpha , x) \,d x= \frac{\curlyvee _{\lambda}(\alpha , \varsigma -\omega )}{(\varsigma -\omega )^{\alpha}}- \frac{\curlyvee _{\lambda}(\alpha +1, \varsigma -\omega )}{(\varsigma -\omega )^{\alpha +1}}$.

Definition 2

(See [18, 22])

A function $\mathcal{F}$ is defined on the interval $[\omega , \varsigma ]$, considering $\mathcal{F}\in L_{1}[\omega , \varsigma ]$, the Riemann–Liouville fractional integral of order $\alpha > 0$, is defined as follows:

$$\begin{aligned} J_{\omega +}^{\alpha}\mathcal{F}(x)=\frac{1}{\Gamma (\alpha )} \int _{ \omega}^{x}(x-\xi )^{\alpha -1}\mathcal{F}(\xi )\,d\xi ,\quad x>\omega \end{aligned}$$

and

$$\begin{aligned} J_{\varsigma -}^{\alpha}\mathcal{F}(x)=\frac{1}{\Gamma (\alpha )} \int _{x}^{\varsigma}(\xi -x)^{\alpha -1}\mathcal{F}(\xi )\,d\xi , \quad x< \varsigma . \end{aligned}$$

Here, $\Gamma (\alpha )$ is the Gamma function and $J_{\omega +}^{0}\mathcal{F}(x)=J_{\varsigma -}^{0}\mathcal{F}(x)= \mathcal{F}(x)$.

Definition 3

(See [24, 26])

Tempered fractional integral operators are defined as:

$$\begin{aligned} \mathcal{J}_{\omega +}^{(\alpha , \lambda )}\mathcal{F}(x)= \frac{1}{\Gamma (\alpha )} \int _{\omega}^{x}(x-\xi )^{\alpha -1}e^{- \lambda (x-\xi )} \mathcal{F}(\xi )\,d\xi ,\quad x\in [\omega , \varsigma ] \end{aligned}$$

and

$$\begin{aligned} \mathcal{J}_{\varsigma -}^{(\alpha , \lambda )}\mathcal{F}(x)= \frac{1}{\Gamma (\alpha )} \int _{x}^{\varsigma}(\xi -x)^{\alpha -1}e^{- \lambda (\xi -x)} \mathcal{F}(\xi )\,d\xi ,\quad x\in [\omega , \varsigma ]. \end{aligned}$$

Here, $\mathcal{F}\in L_{1}[\omega ,\varsigma ]$, $\alpha >0$ and $\lambda \geq 0$.

In the case of $\lambda =0$, 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 Milne-type 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 $\lambda =0$. Under these specified conditions, if we assume $\alpha =1$, the inequalities are reduced to basic Milne-type inequalities.

2 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 $\mathcal{F}:[\omega , \varsigma ] \rightarrow \mathbb{R}$ is an absolutely continuous function on the interval $(\omega , \varsigma )$ with $\mathcal{F'} \in L_{1}[\omega , \varsigma ]$. In this case, the following equality is valid:

$$\begin{aligned} &\frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \\ &\quad = \frac{(\varsigma -\omega )^{\alpha +1}}{2^{\alpha +2} \curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \int _{0}^{1} \biggl(\curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}( \alpha , \xi )+\frac{1}{3}\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr) \\ &\qquad{} \times \biggl[\mathcal{F'} \biggl( \biggl(\frac{1-\xi}{2} \biggr) \omega + \biggl(\frac{1+\xi}{2} \biggr)\varsigma \biggr)- \mathcal{F'} \biggl( \biggl(\frac{1+\xi}{2} \biggr)\omega + \biggl(\frac{1-\xi}{2} \biggr)\varsigma \biggr) \biggr]\,d\xi . \end{aligned}$$

(1)

Proof

Through the utilization of the integration by parts technique, we acquire

$$\begin{aligned} I_{1}& = \int _{0}^{1} \biggl(\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}( \alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr) \biggl[ \mathcal{F'} \biggl( \biggl(\frac{1+\xi}{2} \biggr)\omega + \biggl( \frac{1-\xi}{2} \biggr)\varsigma \biggr) \biggr]\,d\xi \\ &=-\frac{2}{\varsigma -\omega} \biggl[ \biggl(\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3} \curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr) \mathcal{F} \biggl( \biggl(\frac{1+\xi}{2} \biggr)\omega + \biggl( \frac{1-\xi}{2} \biggr)\varsigma \biggr)\bigg|_{0}^{1} \\ &\quad{} +\frac{2}{\varsigma -\omega} \int _{0}^{1}\xi ^{\alpha -1}e^{- \lambda (\frac{\varsigma -\omega}{2})\xi} \mathcal{F'} \biggl( \biggl( \frac{1+\xi}{2} \biggr)\omega + \biggl(\frac{1-\xi}{2} \biggr) \varsigma \biggr)\,d\xi \\ &=-\frac{2}{\varsigma -\omega} \biggl[ \biggl(\frac{4}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr)\mathcal{F}( \omega )+ \biggl( \frac{1}{3}\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr)\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr) \biggr] \\ &\quad{} + \frac{2^{\alpha +1}\Gamma (\alpha )}{(\varsigma -\omega )^{\alpha +1}} \mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr). \end{aligned}$$

(2)

By taking the same steps, we derive

$$\begin{aligned} I_{2}&=\frac{2}{\varsigma -\omega} \biggl[ \biggl( \frac{4}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr)\mathcal{F}( \varsigma )- \biggl(\frac{1}{3}\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr) \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr) \biggr] \\ &\quad{} - \frac{2^{\alpha +1}\Gamma (\alpha )}{(\varsigma -\omega )^{\alpha +1}} \mathcal{J}_{\varsigma -}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr). \end{aligned}$$

(3)

By (2) and (3), this yields

$$\begin{aligned} \frac{(\varsigma -\omega )^{\alpha +1}}{2^{\alpha +2}\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})}[I_{2}-I_{1}]&= \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\quad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr]. \end{aligned}$$

(4)

The proof of Lemma 1 is concluded. □

Theorem 1

Consider the conditions outlined in Lemma 1and the convexity of the function $|\mathcal{F'}|$ on the interval $[\omega , \varsigma ]$, then, we attain the following Milne-type inequalities for tempered fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )- \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(\varsigma -\omega )^{\alpha +1}}{2^{\alpha +2} \curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \mathsf{\Omega _{1}}(\alpha , \lambda ) \bigl[ \bigl\vert \mathcal{F'}(\omega ) \bigr\vert + \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert \bigr], \end{aligned}$$

(5)

where

$$ \begin{aligned} &{\Omega _{1}}(\alpha , \lambda )= \int _{0}^{1} \biggl\vert \curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}(\alpha , \xi )+ \frac{1}{3}\curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \,d\xi . \end{aligned} $$

Proof

By applying the absolute value in Lemma 1 and considering the convexity of $|\mathcal{F'}|$, we acquire

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )- \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{}- \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(\varsigma -\omega )^{\alpha +1}}{2^{\alpha +2}\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \int _{0}^{1} \biggl\vert \curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}( \alpha , \xi )+\frac{1}{3}\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \\ &\qquad{} \times \biggl[ \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1-\xi}{2} \biggr)\omega + \biggl(\frac{1+\xi}{2} \biggr)\varsigma \biggr) \biggr\vert + \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1+\xi}{2} \biggr) \omega + \biggl(\frac{1-\xi}{2} \biggr)\varsigma \biggr) \biggr\vert \biggr]\,d\xi \\ &\quad \leq \frac{(\varsigma -\omega )^{\alpha +1}}{2^{\alpha +2}\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \int _{0}^{1} \biggl\vert \curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}( \alpha , \xi )+\frac{1}{3}\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \\ &\qquad{} \times \biggl[\frac{1+\xi}{2} \bigl\vert \mathcal{F'}( \omega ) \bigr\vert +\frac{1-\xi}{2} \bigl\vert \mathcal{F'}( \varsigma ) \bigr\vert +\frac{1-\xi}{2} \bigl\vert \mathcal{F'}( \omega ) \bigr\vert + \frac{1+\xi}{2} \bigl\vert \mathcal{F'}( \varsigma ) \bigr\vert \biggr]\,d\xi \\ &\quad = \frac{(\varsigma -\omega )^{\alpha +1}}{2^{\alpha +2} \curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \mathsf{\Omega _{1}}(\alpha , \lambda ) \bigl[ \bigl\vert \mathcal{F'}(\omega ) \bigr\vert + \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert \bigr]. \end{aligned}$$

(6)

Consequently, the proof is concluded. □

Remark 2

Assume $\lambda =0$ in Theorem 1, we derive the subsequent Milne’s rule-type inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{2^{\alpha -1}\Gamma (\alpha +1)}{(\varsigma -\omega )^{\alpha}} \biggl[J_{\varsigma -}^{\alpha} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+J_{\omega +}^{\alpha} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{\alpha (\varsigma -\omega )}{4}\mathsf{\Omega _{1}}( \alpha , 0) \bigl[ \bigl\vert \mathcal{F'}(\omega ) \bigr\vert + \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert \bigr], \end{aligned}$$

which is defined in [9, Theorem 1].

Remark 3

When setting $\lambda =0$ and $\alpha =1$ in Theorem 1, we attain the subsequent Milne’s rule-type inequality:

$$\begin{aligned}& \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{1}{\varsigma -\omega} \int _{\omega}^{\varsigma}\mathcal{F}(\xi )\,d \xi \biggr\vert \\& \quad \leq \frac{5(\varsigma -\omega )}{24} \bigl( \bigl\vert \mathcal{F'}( \omega ) \bigr\vert + \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert \bigr), \end{aligned}$$

which is obtained in [7].

Theorem 2

Consider the conditions outlined in Lemma 1and the convexity of the function $|\mathcal{F'}|^{q}, q>1$ on the interval $[\omega , \varsigma ]$. Then, we have the following Milne-type inequalities for tempered fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )- \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{\mathsf{\Phi _{1}^{p}}(\alpha , \lambda )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[ \biggl( \frac{3 \vert \mathcal{F'}(\omega ) \vert ^{q} + \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \frac{ \vert \mathcal{F'}(\omega ) \vert ^{q}+3 \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{\mathsf{4\Phi _{1}^{p}}(\alpha , \lambda )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \bigl[ \bigl\vert \mathcal{F'}(\omega ) \bigr\vert ^{q}+ \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \bigr]. \end{aligned}$$

(7)

Here, $q^{-1}+p^{-1}=1$ and

$$\begin{aligned} &\mathsf{\Phi _{1}^{p}}(\alpha , \lambda )= \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}( \alpha , 1) \biggr\vert ^{p} \biggr)^{\frac{1}{p}}\,d\xi . \end{aligned}$$

Proof

By utilizing Hölder’s inequality in (6), this yields

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(\varsigma -\omega )^{\alpha +1}}{2^{\alpha +2} \curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl\{ \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert ^{p}\,d\xi \biggr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl( \int _{0}^{1} \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1+\xi}{2} \biggr)\omega + \biggl(\frac{1-\xi}{2} \biggr) \varsigma \biggr) \biggr\vert ^{q}\,d\xi \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}( \alpha , \xi )+\frac{1}{3}\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert ^{p}\,d\xi \biggr)^{ \frac{1}{p}} \\ &\qquad{} \times \biggl( \int _{0}^{1} \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1-\xi}{2} \biggr)\omega + \biggl(\frac{1+\xi}{2} \biggr) \varsigma \biggr) \biggr\vert ^{q}\,d\xi \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$

By utilizing the convexity of $|\mathcal{F'}|^{q}$, we attain

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{(\varsigma -\omega )^{\alpha +1}}{2^{\alpha +2}\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl\{ \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert ^{p}\,d\xi \biggr)^{\frac{1}{p}} \\ &\qquad{} \times \biggl[ \biggl( \int _{0}^{1} \biggl[\frac{1+\xi}{2} \bigl\vert \mathcal{F'}(\omega ) \bigr\vert ^{q}+ \frac{1-\xi}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr]\,d\xi \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \int _{0}^{1} \biggl[ \frac{1-\xi}{2} \bigl\vert \mathcal{F'}(\omega ) \bigr\vert ^{q}+ \frac{1+\xi}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr]\,d\xi \biggr)^{\frac{1}{q}} \biggr] \biggr\} \\ &\quad = \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{\mathsf{\Phi _{1}^{p}}(\alpha , \lambda )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[ \biggl( \frac{3 \vert \mathcal{F'}(\omega ) \vert ^{q} + \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \frac{ \vert \mathcal{F'}(\omega ) \vert ^{q}+3 \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$

The first inequality of (7) is proved. For the proof of the second inequality, let $\omega _{1}=$ $3 \vert \mathcal{F}^{\prime} (\omega ) \vert ^{q}, \varsigma _{1}= \vert \mathcal{F}^{\prime} (\varsigma ) \vert ^{q}$, $\omega _{2}= \vert \mathcal{F}^{\prime} (\omega ) \vert ^{q}$, and $\varsigma _{2}=3 \vert \mathcal{F}^{\prime} (\varsigma _{2} ) \vert ^{q}$. Leveraging the provided information that

$$\begin{aligned} \sum_{\kappa =1}^{n} (\omega _{\kappa}+ \varsigma _{\kappa} )^{s} \leq \sum_{\kappa =1}^{n} \omega _{\kappa}^{s}+\sum_{ \kappa =1}^{n} \varsigma _{\kappa}^{s}, \quad 0 \leq s< 1 \end{aligned}$$

and $1+3^{\frac{1}{q}} \leq 4$, the required result can be derived directly. With this, the proof of Theorem 2 is accomplished. □

Remark 4

Setting $\lambda =0$ in Theorem 2, we obtain the subsequent Milne’s rule-type inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{2^{\alpha -1}\Gamma (\alpha +1)}{(\varsigma -\omega )^{\alpha}} \biggl[J_{\varsigma -}^{\alpha} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+J_{\omega +}^{\alpha} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{\alpha (\varsigma -\omega )}{4} \bigl( \mathsf{\Phi _{1}^{p}}( \alpha , 0) \bigr) \biggl[ \biggl( \frac{3 \vert \mathcal{F'}(\omega ) \vert ^{q} + \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}}+ \biggl( \frac{ \vert \mathcal{F'}(\omega ) \vert ^{q}+3 \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq \frac{\alpha (\varsigma -\omega )}{4}\bigl(\mathsf{4\Phi _{1}^{p}}( \alpha , 0)\bigr) \bigl[ \bigl\vert \mathcal{F'}(\omega ) \bigr\vert + \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert \bigr], \end{aligned}$$

which is presented by Budak et al. in [9, Theorem 2].

Remark 5

Assume $\lambda =0$ and $\alpha =1$ in Theorem 2, we deduce the subsequent Milne’s rule-type inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{1}{\varsigma -\omega} \int _{a}^{b}\mathcal{F}(\xi )\,d\xi \biggr\vert \\ &\quad \leq \frac{\varsigma -\omega}{12} \biggl(\frac{4^{p+1}-1}{3(p+1)} \biggr)^{\frac{1}{p}} \biggl[ \biggl( \frac{3 \vert \mathcal{F'}(\omega ) \vert ^{q} + \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}}+ \biggl( \frac{ \vert \mathcal{F'}(\omega ) \vert ^{q}+3 \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{4} \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq \frac{\varsigma -\omega}{12} \biggl(\frac{4^{p+2}-4}{3(p+1)} \biggr)^{\frac{1}{p}} \bigl[ \bigl\vert \mathcal{F'}(\omega ) \bigr\vert + \bigl\vert \mathcal{F'}( \varsigma ) \bigr\vert \bigr], \end{aligned}$$

which is obtained in [9, Corollary 1].

Theorem 3

Consider the conditions outlined in Lemma 1and the convexity of the function $|\mathcal{F'}|^{q}, q\geq 1$ on the interval $[\omega , \varsigma ]$. Then, we have the following Milne-type inequalities for tempered fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )- \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda ) )^{1-\frac{1}{q}}}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl\{ \biggl( \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda ) +\mathsf{\Omega _{2}}(\alpha , \lambda ) )}{2} \bigl\vert \mathcal{F'}(\omega ) \bigr\vert ^{q} \\ &\qquad{} + \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda ) -\mathsf{\Omega _{2}}(\alpha , \lambda ) )}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda )-\mathsf{\Omega _{2}}(\alpha , \lambda ) )}{2} \bigl\vert \mathcal{F'}( \omega ) \bigr\vert ^{q}+ \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda )+\mathsf{\Omega _{2}}(\alpha , \lambda ) )}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr\} , \end{aligned}$$

(8)

where ${\Omega _{1}}(\alpha , \lambda )$ is defined in Theorem 1, and

$$ \begin{aligned} &{\Omega _{2}}(\alpha , \lambda )= \int _{0}^{1} \xi \biggl\vert \curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}(\alpha , \xi )+ \frac{1}{3}\curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \,d\xi . \end{aligned} $$

Proof

By employing the Power mean inequality in (6), we acquire

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{1}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl\{ \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \,d\xi \biggr)^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl[ \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1+\xi}{2} \biggr)\omega + \biggl( \frac{1-\xi}{2} \biggr)\varsigma \biggr) \biggr\vert ^{q}\,d\xi \biggr)^{ \frac{1}{q}} \\ &\qquad{} + \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \\ &\qquad{}\times \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1-\xi}{2} \biggr)\omega + \biggl( \frac{1+\xi}{2} \biggr)\varsigma \biggr) \biggr\vert ^{q}\,d\xi \biggr)^{ \frac{1}{q}} \biggr] \biggr\} . \end{aligned}$$

Considering the convexity of $|\mathcal{F'}|^{q}$ on the interval $[\omega , \varsigma ]$, we derive

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{1}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl\{ \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \,d\xi \biggr)^{1-\frac{1}{q}} \\ &\qquad{} \times \biggl[ \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \\ &\qquad{}\times \biggl( \frac{1+\xi}{2} \bigl\vert \mathcal{F'}( \omega ) \bigr\vert ^{q}+\frac{1-\xi}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr)\,d\xi \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \\ &\qquad{}\times \biggl( \frac{1-\xi}{2} \bigl\vert \mathcal{F'}( \omega ) \bigr\vert ^{q}+\frac{1+\xi}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr)\,d\xi \biggr)^{\frac{1}{q}} \biggr] \biggr\} \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda ) )^{1-\frac{1}{q}}}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl\{ \biggl( \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda ) +\mathsf{\Omega _{2}}(\alpha , \lambda ) )}{2} \bigl\vert \mathcal{F'}(\omega ) \bigr\vert ^{q} \\ &\qquad{} + \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda ) -\mathsf{\Omega _{2}}(\alpha , \lambda ) )}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda )-\mathsf{\Omega _{2}} (\alpha , \lambda ) )}{2} \bigl\vert \mathcal{F'}( \omega ) \bigr\vert ^{q}+ \frac{ (\mathsf{\Omega _{1}}(\alpha , \lambda ) +\mathsf{\Omega _{2}}(\alpha , \lambda ) )}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr\} . \end{aligned}$$

This concludes the proof. □

Remark 6

Taking $\lambda =0$ in Theorem 3, we acquire the subsequent Milne’s rule-type inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{2^{\alpha -1}\Gamma (\alpha +1)}{(\varsigma -\omega )^{\alpha}} \biggl[J_{\varsigma -}^{\alpha} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+J_{\omega +}^{\alpha} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{\alpha (\varsigma -\omega )}{4} \bigl( \mathsf{\Omega _{1}}(\alpha , 0) \bigr)^{1-\frac{1}{q}} \biggl\{ \biggl( \frac{ (\mathsf{\Omega _{1}}(\alpha , 0) +\mathsf{\Omega _{2}}(\alpha , 0) )}{2} \bigl\vert \mathcal{F'}(\omega ) \bigr\vert ^{q} \\ &\qquad{} + \frac{ (\mathsf{\Omega _{1}}(\alpha , 0)-\mathsf{\Omega _{2}}(\alpha , 0) )}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \frac{ (\mathsf{\Omega _{1}}(\alpha , 0)-\mathsf{\Omega _{2}}(\alpha , 0) )}{2} \bigl\vert \mathcal{F'}( \omega ) \bigr\vert ^{q}+ \frac{ (\mathsf{\Omega _{1}}(\alpha , 0)+\mathsf{\Omega _{2}}(\alpha , 0) )}{2} \bigl\vert \mathcal{F'}(\varsigma ) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr\} , \end{aligned}$$

which is found in [9, Theorem 3].

Remark 7

If we take $\lambda =0$ and $\alpha =1$ in Theorem 3, we deduce the subsequent Milne’s rule-type inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{1}{\varsigma -\omega} \int _{\omega}^{\varsigma}\mathcal{F}(\xi )\,d \xi \biggr\vert \\ &\quad \leq \frac{5(\varsigma -\omega )}{24} \biggl[ \biggl( \frac{4 \vert \mathcal{F'}(\omega ) \vert ^{q}+ \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{5} \biggr)^{\frac{1}{q}}+ \biggl( \frac{ \vert \mathcal{F'}(\omega ) \vert ^{q}+4 \vert \mathcal{F'}(\varsigma ) \vert ^{q}}{5} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$

which is obtained in [7, Remark 3.2].

Theorem 4

Consider the conditions outlined in Lemma 1to hold. If there exist $m, M\in \mathbb{R}$ such that $m\leq \mathcal{F'}(\xi )\leq M$ for $\xi \in [\omega , \varsigma ]$, then, we obtain the following Milne-type inequalities for tempered fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )- \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{\mathsf{\Omega _{1}}(\alpha , \lambda )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})}(M-m), \end{aligned}$$

(9)

where ${\Omega _{1}}(\alpha , \lambda )$ is defined as in Theorem 1.

Proof

From Lemma 1, it is easy to write

$$\begin{aligned} &\frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{1}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl\{ \int _{0}^{1} \biggl(\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}( \alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr) \\ &\qquad{} \times \biggl[\mathcal{F'} \biggl( \biggl( \frac{1-\xi}{2} \biggr) \omega + \biggl(\frac{1+\xi}{2} \biggr)\varsigma \biggr)- \frac{m+M}{2} \biggr]\,d\xi \\ &\qquad{} + \int _{0}^{1} \biggl(\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}( \alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr) \\ &\qquad{} \times \biggl[\frac{m+m}{2}-\mathcal{F'} \biggl( \biggl( \frac{1+\xi}{2} \biggr)\omega + \biggl(\frac{1-\xi}{2} \biggr) \varsigma \biggr) \biggr]\,d\xi \biggr\} . \end{aligned}$$

(10)

By employing the properties of modulus in equation (10), we can derive

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{1}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl\{ \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \\ &\qquad{} \times \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1-\xi}{2} \biggr) \omega + \biggl(\frac{1+\xi}{2} \biggr)\varsigma \biggr)- \frac{m+M}{2} \biggr\vert \,d\xi \\ &\qquad{} + \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \\ &\qquad{} \times \biggl\vert \frac{m+m}{2}-\mathcal{F'} \biggl( \biggl( \frac{1+\xi}{2} \biggr)\omega + \biggl(\frac{1-\xi}{2} \biggr) \varsigma \biggr) \biggr\vert \,d\xi \biggr\} . \end{aligned}$$

Based on the given assumption $m\leq \mathcal{F'}(\xi )\leq M$ for $\xi \in [\omega , \varsigma ]$, this yields

$$\begin{aligned} \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1-\xi}{2} \biggr)\omega + \biggl(\frac{1+\xi}{2} \biggr)\varsigma \biggr)-\frac{m+M}{2} \biggr\vert \leq \frac{M-m}{2} \end{aligned}$$

(11)

and

$$\begin{aligned} \biggl\vert \frac{m+m}{2}-\mathcal{F'} \biggl( \biggl(\frac{1+\xi}{2} \biggr) \omega + \biggl(\frac{1-\xi}{2} \biggr)\varsigma \biggr) \biggr\vert \leq \frac{M-m}{2}. \end{aligned}$$

(12)

With the utilization of inequalities (11) and (12), we achieve

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad {}- \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{1}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[ \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \,d\xi \biggr](M-m) \\ &\quad = \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{\mathsf{\Omega _{1}}(\alpha , \lambda )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})}(M-m). \end{aligned}$$

Hence, the proof is effectively concluded. □

Corollary 1

Under the conditions of Theorem 4, if there exist $M\in \mathbb{R}^{+}$ such that $|\mathcal{F'}(\xi )|\leq M$, for all $\xi \in [\omega , \varsigma ]$, then we attain

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{\mathsf{\Omega _{1}}(\alpha , \lambda )}{\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})}M. \end{aligned}$$

Remark 8

If we set $\lambda =0$, in Corollary 1, we have

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{2^{\alpha -1}\Gamma (\alpha +1)}{(\varsigma -\omega )^{\alpha}} \biggl[J_{\omega +}^{\alpha} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+J_{\varsigma -}^{\alpha} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{\alpha (\varsigma -\omega )}{2}\mathsf{\Omega _{1}}( \alpha , 0)M, \end{aligned}$$

which is obtained in [9, Corollary 3].

Remark 9

Putting $\lambda =0$ and $\alpha =1$ in Corollary 1, this yields

$$\begin{aligned} \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{1}{\varsigma -\omega} \int _{\omega}^{\varsigma}\mathcal{F}(\xi )\,d \xi \biggr\vert \leq \frac{5(\varsigma -\omega )}{12}M, \end{aligned}$$

which was obtained by Alomari and Liu [3].

Remark 10

Taking $\lambda =0$ in Theorem 4, we attain

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{\Gamma (\alpha +1)}{2(\varsigma -\omega )^{\alpha}} \bigl[J_{ \varsigma -}^{\alpha} \mathcal{F}(\omega )+J_{\omega +}^{\alpha} \mathcal{F}(\varsigma ) \bigr] \biggr\vert \\ &\quad \leq \frac{\alpha (\varsigma -\omega )}{4}\mathsf{\Omega _{1}}( \alpha , 0) (M-m), \end{aligned}$$

which is presented in [9, Theorem 4].

Remark 11

Assume $\lambda =0$ and $\alpha =1$ in Theorem 4, we deduce the subsequent inequality:

$$\begin{aligned} \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{1}{\varsigma -\omega} \int _{\omega}^{\varsigma}\mathcal{F}(\xi )\,d \xi \biggr\vert \leq \frac{5(\varsigma -\omega )}{24} (M-m ), \end{aligned}$$

which is defined in [9, Corollary 2].

Theorem 5

Consider the conditions outlined in Lemma 1to hold. If $\mathcal{F'}$ is an L-Lipschitzian function on the interval $[\omega , \varsigma ]$, then we obtain the following Milne-type inequalities for tempered fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )- \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +2} \frac{\mathsf{\Omega _{2}}(\alpha , \lambda )}{\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \mathsf{L}. \end{aligned}$$

(13)

Here, ${\Omega _{2}}(\alpha , \lambda )$ is defined as in Theorem 3.

Proof

Utilizing the modulus in Lemma 1, we attain

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \varsigma -\omega )} \bigl[\mathcal{J}_{\varsigma -}^{(\alpha , \lambda )} \mathcal{F}( \omega )+\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F}( \varsigma ) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{1}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[ \int _{0}^{1} \biggl\vert \curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , \xi )+\frac{1}{3}\curlyvee _{ \lambda (\frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \\ &\qquad{} \times \biggl\vert \mathcal{F'} \biggl( \biggl( \frac{1-\xi}{2} \biggr)\omega + \biggl(\frac{1+\xi}{2} \biggr)\varsigma \biggr)- \mathcal{F'} \biggl( \biggl(\frac{1+\xi}{2} \biggr)\omega + \biggl( \frac{1-\xi}{2} \biggr)\varsigma \biggr) \biggr\vert \,d\xi \biggr]. \end{aligned}$$

Since $|\mathcal{F'}|$ is an L-Lipschitzian function, we can conclude

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \varsigma -\omega )} \bigl[\mathcal{J}_{\varsigma -}^{(\alpha , \lambda )} \mathcal{F}( \omega )+\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F}( \varsigma ) \bigr] \biggr\vert \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +1} \frac{1}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \int _{0}^{1} \biggl\vert \curlyvee _{\lambda (\frac{\varsigma -\omega}{2})}( \alpha , \xi )+\frac{1}{3}\curlyvee _{\lambda ( \frac{\varsigma -\omega}{2})}(\alpha , 1) \biggr\vert \mathsf{L}( \varsigma -\omega )\xi \,d\xi \\ &\quad \leq \biggl(\frac{\varsigma -\omega}{2} \biggr)^{\alpha +2} \frac{\mathsf{\Omega _{2}}(\alpha , \lambda )}{\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \mathsf{L}. \end{aligned}$$

Consequently, the demonstration is concluded. □

Remark 12

By setting $\lambda =0$, in Theorem 5, we attain

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{\Gamma (\alpha +1)}{2(\varsigma -\omega )^{\alpha}} \bigl[J_{ \varsigma -}^{\alpha} \mathcal{F}(\omega )+J_{\omega +}^{\alpha} \mathcal{F}(\varsigma ) \bigr] \biggr\vert \\ &\quad \leq \frac{\alpha (\varsigma -\omega )^{2}}{4}\mathsf{\Omega _{2}}( \alpha , 0)\mathsf{L}, \end{aligned}$$

which is defined in [9, Theorem 5].

Remark 13

By setting $\lambda =0$ and $\alpha =1$ in Theorem 5, this yields

$$\begin{aligned} \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{1}{\varsigma -\omega} \int _{\omega}^{\varsigma}\mathcal{F}(\xi )\,d \xi \biggr\vert \leq \frac{(\varsigma -\omega )^{2}}{8}\mathsf{L}, \end{aligned}$$

which is presented in [9, Corollary 4].

Theorem 6

Assume $\mathcal{F}:[\omega , \varsigma ]\to \mathbb{R}$ exhibits a bounded variation over the interval $[\omega , \varsigma ]$. Then, we have the following Milne-type inequality that is specifically tailored for tempered fractional integrals:

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )- \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{1}{2\curlyvee _{\lambda }(\alpha ,\frac{\varsigma -\omega}{2})} \biggl[\max \biggl\{ \frac{\curlyvee _{\lambda }(\alpha , \frac{\varsigma -\omega}{2})}{3}, \biggl\vert \curlyvee _{\lambda } \biggl( \alpha , \frac{\varsigma -\omega}{2} \biggr)+ \frac{\curlyvee _{\lambda }(\alpha , \frac{\varsigma -\omega}{2})}{3} \biggr\vert \biggr\} \biggr] \bigvee _{\omega }^{\varsigma }( \mathcal{F}), \end{aligned}$$

(14)

where $\bigvee_{\omega}^{\varsigma}(\mathcal{F})$ indicate the total variation of $\mathcal{F}$ on $[\omega , \varsigma ]$.

Proof

Define the mappings

$$\begin{aligned} \mathsf{T}_{\alpha}(\varkappa )=\textstyle\begin{cases} {-}\curlyvee _{\lambda}(\alpha , \frac{\omega +\varsigma}{2}-\varkappa )- \frac{\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})}{3}, & \omega \leq \varkappa \leq \frac{\omega +\varsigma}{2} \\ \curlyvee _{\lambda}(\alpha , \varkappa -\frac{\omega +\varsigma}{2})+ \frac{\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})}{3} & \frac{\omega +\varsigma}{2}< \varkappa \leq \varsigma . \end{cases}\displaystyle \end{aligned}$$

By employing the integration by parts technique, we acquire

$$\begin{aligned} & \int _{\omega}^{\varsigma} \mathsf{T}_{\alpha}(\varkappa )\,d \mathcal{F}(\varkappa ) \\ &\quad = \int _{\omega}^{\frac{\omega +\varsigma}{2}} \biggl({-}\curlyvee _{ \lambda} \biggl(\alpha , \frac{\omega +\varsigma}{2}-\varkappa \biggr) - \frac{\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggr) \,d\mathcal{F}(\varkappa ) \\ &\qquad{} + \int _{\frac{\omega +\varsigma}{2}}^{ \varsigma} \biggl(\curlyvee _{\lambda} \biggl(\alpha , \varkappa - \frac{\omega +\varsigma}{2} \biggr) + \frac{\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggr)\,d \mathcal{F}(\varkappa ) \\ &\quad = \biggl(-\curlyvee _{\lambda} \biggl(\alpha , \frac{\omega +\varsigma}{2}- \varkappa \biggr)- \frac{\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggr)\mathcal{F}(\varkappa )\bigg|_{\omega}^{ \frac{\omega +\varsigma}{2}} \\ &\qquad{} - \int _{\omega}^{ \frac{\omega +\varsigma}{2}} \biggl(\frac{\omega +\varsigma}{2}- \varkappa \biggr)^{\alpha -1}e^{-\lambda ( \frac{\omega +\varsigma}{2}-\varkappa )}\mathcal{F}(\varkappa )\,d \mathcal{F}( \varkappa ) \\ &\qquad{} + \biggl(\curlyvee _{\lambda} \biggl(\alpha , \varkappa - \frac{\omega +\varsigma}{2} \biggr)+ \frac{\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggr)\mathcal{F}(\varkappa )\bigg|_{\frac{\omega +\varsigma}{2}}^{ \varsigma} \\ &\qquad{} - \int _{\frac{\omega +\varsigma}{2}}^{\varsigma} \biggl( \varkappa -\frac{\omega +\varsigma}{2} \biggr)^{\alpha -1}e^{- \lambda (\varkappa -\frac{\omega +\varsigma}{2} )} \mathcal{F}(\varkappa )\,d\mathcal{F}( \varkappa ) \\ &\quad =- \frac{\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr)+ \frac{4\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \mathcal{F}(\omega )-{\Gamma (\alpha )}\mathcal{J}_{\omega +}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \\ &\qquad{} + \frac{4\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \mathcal{F}(\varsigma )- \frac{\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr)-{\Gamma ( \alpha )} \mathcal{J}_{\varsigma -}^{(\alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \\ &\quad = \frac{2\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} -{ \Gamma (\alpha )} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{ \varsigma -}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr) \biggr]. \end{aligned}$$

That is,

$$\begin{aligned} &\frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \\ &\quad = \frac{1}{2\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )} \int _{\omega}^{\varsigma} \mathsf{T}_{\alpha}(\varkappa )\,d \mathcal{F}(\varkappa ). \end{aligned}$$

It is a well-established fact that if $\mathsf{g}, \mathcal{F}:[\omega , \varsigma ] \rightarrow \mathbb{R}$ satisfy the conditions where g is continuous on $[\omega , \varsigma ]$ and $\mathcal{F}$ is of bounded variation on $[\omega , \varsigma ]$, then $\int _{\omega}^{\varsigma }\mathsf{g}(\xi ) \,d\mathcal{F}(\xi )$ exists and

$$\begin{aligned} \biggl\vert \int _{\omega}^{\varsigma} \mathsf{g}(\xi ) \,d\mathcal{F}(\xi ) \biggr\vert \leq \sup_{\xi \in [\omega , \varsigma ]} \bigl\vert \mathsf{g}(\xi ) \bigr\vert \bigvee_{\omega}^{\varsigma}(\mathcal{F}). \end{aligned}$$

(15)

However, employing (15), we have

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2\mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr] \\ &\qquad{} - \frac{\Gamma (\alpha )}{2\curlyvee _{\lambda}(\alpha , \frac{\varsigma -\omega}{2})} \biggl[\mathcal{J}_{\omega +}^{(\alpha , \lambda )} \mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+\mathcal{J}_{\varsigma -}^{( \alpha , \lambda )} \mathcal{F} \biggl(\frac{\omega +\varsigma}{2} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{1}{2\curlyvee _{\lambda }(\alpha , \frac{\varsigma -\omega}{2})} \biggl\vert \int _{\omega }^{\varsigma }\mathsf{T}_{\alpha }( \varkappa )\,d \mathcal{F}(\varkappa ) \biggr\vert \\ &\quad \leq \frac{1}{2\curlyvee _{\lambda }(\alpha , \frac{\varsigma -\omega }{2})} \biggl[ \biggl\vert \int _{\omega }^{\frac{\omega +\varsigma }{2}} \biggl(\curlyvee _{\lambda} \biggl(\alpha , \frac{\omega +\varsigma}{2}-\varkappa \biggr) + \frac{\curlyvee _{\lambda} (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggr) \,d \mathcal{F}(\varkappa ) \biggr\vert \\ &\qquad{} + \biggl\vert \int _{\frac{\omega +\varsigma }{2}}^{ \varsigma } \biggl( \curlyvee _{\lambda } \biggl(\alpha ,\varkappa - \frac{\omega +\varsigma}{2} \biggr) + \frac{\curlyvee _{\lambda } (\alpha ,\frac{\varsigma -\omega}{2} )}{3} \biggr) \,d \mathcal{F}(\varkappa ) \biggr\vert \biggr] \\ &\quad \leq \frac{1}{2\curlyvee _{\lambda }(\alpha , \frac{\varsigma -\omega }{2})} \Biggl[ \sup_{\varkappa \in [ \omega , \frac{\omega +\varsigma }{2} ] } \biggl\vert \curlyvee _{ \lambda } \biggl(\alpha , \frac{\omega +\varsigma}{2}-\varkappa \biggr)+ \frac{\curlyvee _{\lambda } (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggr\vert \bigvee_{\omega }^{\frac{\omega +\varsigma }{2}}( \mathcal{F}) \\ &\qquad{} +\sup_{\varkappa \in [ \frac{\omega +\varsigma }{2}, \varsigma ] } \biggl\vert \curlyvee _{\lambda } \biggl(\alpha , \varkappa -\frac{\omega +\varsigma}{2} \biggr) + \frac{\curlyvee _{\lambda } (\alpha ,\frac{\varsigma -\omega}{2} )}{3} \biggr\vert \bigvee_{\frac{\omega +\varsigma }{2}}^{\varsigma }(\mathcal{F}) \Biggr] \\ &\quad \leq \frac{1}{2\curlyvee _{\lambda }(\alpha , \frac{\varsigma -\omega }{2})} \Biggl[ \max \Biggl\{ \frac{\curlyvee _{\lambda } (\alpha , \frac{\varsigma -\omega}{2} )}{3}, \biggl\vert \curlyvee _{\lambda } \biggl(\alpha , \frac{\varsigma -\omega}{2} \biggr)+ \frac{\curlyvee _{\lambda } (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggr\vert \bigvee_{\omega }^{\frac{\omega +\varsigma }{2}}( \mathcal{F}) \Biggr\} \\ & \qquad{} +\max \Biggl\{ \frac{\curlyvee _{\lambda } (\alpha , \frac{\varsigma -\omega}{2} )}{3}, \biggl\vert \curlyvee _{\lambda } \biggl(\alpha , \frac{\varsigma -\omega}{2} \biggr)+ \frac{\curlyvee _{\lambda } (\alpha , \frac{\varsigma -\omega}{2} )}{3} \biggr\vert \bigvee_{\frac{\omega +\varsigma }{2}}^{\varsigma }(\mathcal{F}) \Biggr\} \Biggr] \\ & \quad \leq \frac{1}{2\curlyvee _{\lambda }(\alpha ,\frac{\varsigma -\omega}{2})} \biggl[\max \biggl\{ \frac{\curlyvee _{\lambda }(\alpha , \frac{\varsigma -\omega}{2})}{3}, \biggl\vert \curlyvee _{\lambda } \biggl( \alpha , \frac{\varsigma -\omega}{2} \biggr)+ \frac{\curlyvee _{\lambda }(\alpha , \frac{\varsigma -\omega}{2})}{3} \biggr\vert \biggr\} \biggr] \bigvee _{\omega }^{\varsigma }( \mathcal{F}). \end{aligned}$$

Hence, the proof is finalized. □

Remark 14

Setting $\lambda =0$, in Theorem 6, we find

$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{\Gamma (\alpha +1)}{2(\varsigma -\omega )^{\alpha}} \bigl[J_{ \varsigma -}^{\alpha} \mathcal{F}(\omega )+J_{\omega +}^{\alpha} \mathcal{F}(\varsigma ) \bigr] \biggr\vert \leq \frac{2}{3} \bigvee_{ \omega}^{\varsigma}( \mathcal{F}), \end{aligned}$$

which is obtained in [9, Theorem 6].

Remark 15

Assume $\lambda =0$ and $\alpha =1$ in Theorem 6, this yields

$$\begin{aligned} \biggl\vert \frac{1}{3} \biggl[2 \mathcal{F}(\omega )-\mathcal{F} \biggl( \frac{\omega +\varsigma}{2} \biggr)+2 \mathcal{F}(\varsigma ) \biggr]- \frac{1}{\varsigma -\omega} \int _{\omega}^{\varsigma}\mathcal{F}(\xi )\,d \xi \biggr\vert \leq \frac{2}{3} \bigvee_{\omega}^{\varsigma}( \mathcal{F}), \end{aligned}$$

which is proposed in [3].

3 Conclusions

In summary, our investigation yields new Milne-type inequalities using tempered fractional integrals. We employ essential function classes, to establish a fundamental identity, and derived novel fractional Milne inequalities, extending formulations for tempered fractional integrals. Our findings contribute insights for future studies, generalize prior work, and suggest potential applications in diverse mathematical contexts. The insights and methodologies revealed by our findings concerning Milne-type inequalities through tempered fractional integrals have the potential to serve as a foundation for future explorations in this domain. Readers can further expand on our findings by applying a different kind of fractional integral.