1 Introduction

Interval computation (also known as interval arithmetic or interval analysis) is a powerful numerical tool which allows us to solve uncertain or nonlinear problems in a guaranteed way. It has a long history which can be traced back to Archimedes’ computation of the circumference of a circle. But the systematic study of interval computation only began around the 1920 s. In 1924, Burkill [9] developed some elementary properties of functions of intervals. Shortly afterwards, Kolmogorov [23] generalized Burkill’s results from single-valued functions to multi-valued functions. Of course, there are many other excellent results that have been achieved over the next two decades. Until 1966, the first monograph on interval calculation was published by Moore [32]. Nowadays, interval computation has developed into a multifaceted branch of mathematics with applications to global optimization, robotics, structural engineering, computer graphics, electrical engineering, and many other areas (see [1,2,3, 16, 19]).

Recently, interval computation has gradually been used to deal with a large class of problems involving equations or inequalities, which can be non-smooth or generalized convex. Some classical real-valued inequalities have been considered for extension to interval-valued functions (IVFs) or fuzzy IVFs (see [7, 10, 13, 14, 21, 22]). Among the many types of inequalities, the Hermite–Hadamard (H–H) inequality has attracted much attention. It states that if \(f:I\rightarrow \mathbb {R}\) is a convex function on the interval I of real numbers and \(j_{1},j_{2}\in I\) with \(j_{1}<j_{2}\), then

$$\begin{aligned} \begin{aligned}{} & {} f\Big (\frac{j_{1}+j_{2}}{2}\Big )\le \frac{1}{j_{2}-j_{1}}\int _{j_{1}}^{j_{2}} f(x)\,dx \le \frac{f(j_{1})+f(j_{2})}{2}. \end{aligned} \end{aligned}$$
(1)

The H–H inequality is a valuable tool in the theory of convex functions, providing a two-sided estimate for the mean value of a convex function. It has been developed for different classes of convexity, such as harmonically convex [6, 18], log-preinvex [34], \(\hbar\)-convex [35], and especially for \(\hbar\)-preinvex [28]. Since 2014, various generalizations of H–H inequalities for \(\hbar\)-preinvex functions (\(\hbar\)-PFs) have been established by Latif et al. [24, 25], Matłoka [29, 30], Noor et al. [36], Sun [45], and others. What is more, several H–H-type inequalities have been used to establish bounds and estimates for the integrals of IVFs. These results have implications in the study of fractional calculus and fractional integral operators, as well as in the analysis of functions with interval-valued outputs (see Du et al. [15], Budak et al. [8], Khan et al. [20], Sharma et al. [42], Srivastava et al. [43, 44], and Zhao et al. [49,50,51,52,53]).

Moreover, H–H-type inequalities for IVFs have found applications in optimization theory, economics and so on. In optimization theory, these inequalities have been employed to derive necessary and sufficient conditions for the convexity of optimization problems involving IVFs. These conditions help in formulating efficient algorithms and decision-making processes in real-world optimization problems, where uncertainty or imprecision is present [4, 17, 48]. By considering IVFs, which capture uncertainty or imprecision in economic models, the H–H-type inequalities provide valuable insights into the behavior and properties of economic variables. They contribute to the development of robust decision-making frameworks and risk analysis methods [11, 12, 26, 46]. Overall, the literature demonstrates the significance and relevance of H–H-type inequalities for IVFs in various scientific disciplines. They provide powerful tools for analyzing and dealing with uncertainties, imprecisions, and variations in mathematical models and real-world applications.

Motivated by the above results, we establish some H–H-type inequalities for \(\hbar\)-PIVFs via Riemann–Liouville (R–L) fractional integral, and give accurate proofs for the main theorems originally derived by Srivastava et al. in [43]. In addition, we illustrate our findings through a practical example to demonstrate the validity of our results. Our results generalize previous inequalities presented by [5, 37, 38, 40, 41, 50], and will provide a deeper understanding of the properties of IVFs.

The paper is organized as follows. Section 2 contains some necessary preliminaries. In Sect. 3, we establish some H–H-type inequalities for \(\hbar\)-PIVFs using the R–L fractional integral and give a corresponding example. We end with Sect. 4 of conclusions.

2 Preliminaries

We define an interval \(\varvec{I}\) by

$$\begin{aligned}{} & {} \varvec{I}=[\underline{\varvec{I}},\overline{\varvec{I}}] =\{x\in \mathbb {R}|\ \underline{\varvec{I}}\le x\le \overline{\varvec{I}}\}, \end{aligned}$$

where \(\underline{\varvec{I}}\le \overline{\varvec{I}}\). We writer \(len(\varvec{I})=\overline{\varvec{I}}-\underline{\varvec{I}}\). If \(len(\varvec{I})=0\), then \(\varvec{I}\) is called degenerate. In this paper, all considered intervals will mean non-degenerate intervals. \(\varvec{I}\) is called positive (negative) if \(\underline{\varvec{I}}>0\) (\(\overline{\varvec{I}}<0\)). Let \(\mathbb {R}_{0}^{+}\), \(\mathbb {R}_{I}\) and \(\mathbb {R}_{I} ^{+}\) be sets of all non-negative numbers, intervals and positive intervals of \(\mathbb {R}\), respectively. The partial order “\(\subseteq\)” is defined by

$$\begin{aligned}{} & {} [\underline{ \alpha },\overline{\alpha }]\subseteq [\underline{\beta },\overline{\beta }] \Longleftrightarrow \quad \underline{ \beta }\le \underline{\alpha }, \overline{\alpha }\le \overline{\beta }. \end{aligned}$$
(2)

For \(\eta \in \mathbb {R}\) and \(\alpha \in \mathbb {R}_{I}\), \(\eta \alpha\) is defined by

$$\ \eta \alpha = \eta [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } ,\bar{\alpha }] = \left\{ {\begin{array}{*{20}c} {[\eta \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } ,\eta \bar{\alpha }],} & {{\text{if}}\;\eta \ge 0,} \\ {[\eta \bar{\alpha },\eta \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\alpha } ],} & {{\text{if }}\;\eta < 0.} \\ \end{array} } \right.$$
(3)

For arbitrary \(\alpha , \beta \in \mathbb {R}_{I}\), the four arithmetic operators \((+, -, \cdot , / )\) are defined by

$$\begin{aligned} \begin{aligned}&\alpha +\beta =[\underline{ \alpha },\overline{\alpha }] +[\underline{\beta },\overline{\beta }]=[\underline{\alpha } +\underline{\beta },\overline{\alpha }+\overline{\beta }],\\&\alpha -\beta =[\underline{ \alpha },\overline{\alpha }] -[\underline{\beta },\overline{\beta }] =[\underline{\alpha }-\overline{\beta },\overline{\alpha }-\underline{\beta }],\\&\alpha \cdot \beta =[\min \left\{ \underline{\alpha \beta }, \underline{\alpha }\overline{\beta },\right. \left. \overline{\alpha }\underline{\beta },\overline{\alpha \beta }\right\} , \max \left\{ \underline{\alpha \beta }, \right. \left. \underline{\alpha }\overline{\beta },\overline{\alpha }\underline{\beta },\overline{\alpha \beta }\right\} ],\\&\beta /\alpha = \big [\min \left\{ \underline{\beta }/ \underline{\alpha },\right. \underline{\beta }/ \overline{\alpha },\overline{\beta }/ \left. \underline{\alpha }, \overline{\beta }/ \overline{\alpha }\right\} , \max \left\{ \underline{\beta }/ \underline{\alpha }, \underline{\beta }/ \overline{\alpha },\right. \left. \overline{\beta }/ \underline{\alpha }, \overline{\beta }/ \overline{\alpha }\right\} \big ], (0\notin [\underline{\alpha }, \overline{\alpha }] ). \end{aligned} \end{aligned}$$
(4)

For more details on interval arithmetic, see [33].

Definition 2.1

([47]) A set \(I \subseteq \mathbb {R}^{n}\) is said to be invex with respect to \(\xi :I \times I \rightarrow \mathbb {R}^{n}\), if

$$\begin{aligned} j_{2}+t\xi (j_{1},j_{2})\in I, \qquad (\forall j_{1}, j_{2}\in I, t\in [0,1]). \end{aligned}$$
(5)

Note that, every convex set is invex with respect to \(\xi (j_{1},j_{2})=j_{1}-j_{2}\), but the converse is not true.

Definition 2.2

([47]) Let \(I \subseteq \mathbb {R} ^{n}\) be an invex set with respect to \(\xi :I \times I \rightarrow \mathbb {R}^{n}\). Then, \(\Upsilon :I\rightarrow \mathbb {R}^{n}\) is said to be preinvex with respect to \(\xi\), if

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big ) \le t\Upsilon (j_{1})+(1-t)\Upsilon (j_{2}),\\&\quad (\forall j_{1},j_{2} \in I,t\in [0,1]). \end{aligned} \end{aligned}$$
(6)

The family of all preinvex functions with respect to \(\xi\) on I are denoted by \(S(P, I,\mathbb {R}^{n})\).

In 2021, Sharma et al. [42] gave the definition of \(\hbar\)-PIVFs.

Definition 2.3

([42]) Let \(\hbar :(0,1)\subseteq [ a, b]\rightarrow \mathbb {R}_{0}^{+}\) and \(\hbar \not \equiv 0\). We say that \(\Upsilon :I\rightarrow \mathbb {R}_{I}^{+}\) is a \(\hbar\)-PIVF with respect to \(\xi\), if

$$\begin{aligned}{} & {} \Upsilon \big (y+t\xi (x,y)\big )\supseteq \hbar (t)\Upsilon (x) +\hbar (1-t)\Upsilon (y), \qquad (\forall x,y\in I, t\in [0,1] ). \end{aligned}$$
(7)

Let \(S(\hbar P, I, \mathbb {R}_{I}^{+})\) and \(S(\hbar P, I,\mathbb {R} )\) denote the sets of all \(\hbar\)-PIVFs and \(\hbar\)-preinvex functions with respect to \(\xi\) on I, respectively.

For the further reasoning, we also need the well-known Condition C.

Condition C. ([31]) Let \(I\subseteq \mathbb {R}\) be an invex set with respect to \(\xi :I\times I \rightarrow \mathbb {R}\). We say that \(\xi\) satisfies the Condition C provided for any \(j_{1},j_{2} \in I\) and \(t\in [0,1]\),

$$\begin{aligned} \begin{aligned} \xi \big (j_{2},j_{2} +t\xi (j_{1},j_{2})\big )=-t\xi (j_{1},j_{2}), \end{aligned} \end{aligned}$$
(8)

and

$$\begin{aligned} \qquad \xi \big (j_{1},j_{2} +t\xi (j_{1},j_{2})\big )=(1-t)\xi (j_{1},j_{2}). \end{aligned}$$
(9)

From Condition C, we also have

$$\begin{aligned}{} & {} \xi \big (j_{2} +t_{2}\xi (j_{1},j_{2}),j_{2} +t_{1}\xi (j_{1},j_{2})\big )=(t_{2}-t_{1})\xi (j_{1},j_{2}). \end{aligned}$$
(10)

Let \(\Upsilon :I\rightarrow \mathbb {R}_{I}^{+}\), \(\underline{\Upsilon }\) and \(\overline{\Upsilon }\) are measurable and Lebesgue integrable on \([j_{1}, j_{2}]\). Then, we define \(\int _{j_{1}}^{j_{2}} \Upsilon (x) \,dx\) by

$$\begin{aligned}{} & {} \int _{j_{1}}^{j_{2}} \Upsilon (x) \,dx =\bigg [\int _{j_{1}}^{j_{2}} \underline{\Upsilon } (x) \,dx, \int _{j_{1}}^{j_{2}} \overline{\Upsilon } (x) \,dx \bigg ], \end{aligned}$$
(11)

and we say that \(\Upsilon\) is interval Lebesgue integrable on \([j_{1}, j_{2}]\)(or that \(\Upsilon \in IL_{[j_{1}, j_{2}]}\)).

Definition 2.4

([39]) Let \(\Upsilon \in L[j_{1},j_{2}]\). The left and right R–L fractional integrals \(\mathfrak {J} ^{\alpha }_{(j_{1})^{+}}\) and \(\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\) of order \(\alpha >0\) are defined by

$$\begin{aligned}{} & {} \mathfrak {J} ^{\alpha }_{(j_{1})^{+}}\Upsilon (\omega ) =\frac{1}{\Gamma (\alpha )}\int _{j_{1}}^{\omega }(\omega -x)^{\alpha -1}\Upsilon (x) \,dx, (0 \le j_{1}<\omega \le j_{2} ), \end{aligned}$$
(12)

and

$$\begin{aligned}{} & {} \mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon (\omega ) =\frac{1}{\Gamma (\alpha )}\int _{\omega }^{j_{2}}(x-\omega )^{\alpha -1}\Upsilon (x) \,dx, (0 \le j_{1}\le \omega < j_{2}), \end{aligned}$$
(13)

respectively, where \(\Gamma (\alpha )\) is the Euler Gamma mapping with \(\Gamma (\alpha )=\int _{0}^{\infty } x^{\alpha -1}e^{-x} \,dx\). Note that \(\mathfrak {J} ^{0}_{(j_{1})^{+}}\Upsilon (\omega )= \mathfrak {J} ^{0 }_{(j_{2})^{-}}\Upsilon (\omega )=\Upsilon (\omega )\).

Definition 2.5

([27]) Let \(\Upsilon \in IL_{[j_{1},j_{2}]}\). The left and right interval-valued R–L fractional integrals \(\mathfrak {J} ^{\alpha }_{(j_{1})^{+}}\) and \(\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\) of order \(\alpha >0\) are defined by

$$\begin{aligned}{} & {} \mathfrak {J} ^{\alpha }_{(j_{1})^{+}}\Upsilon (\omega ) =\frac{1}{\Gamma (\alpha )} \int _{j_{1}}^{\omega }(\omega -x)^{\alpha -1}\Upsilon (x) \,dx, (0 \le j_{1}<\omega \le j_{2}), \end{aligned}$$
(14)

and

$$\begin{aligned}{} & {} \mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon (\omega ) =\frac{1}{\Gamma (\alpha )} \int _{\omega }^{j_{2}}(x-\omega )^{\alpha -1}\Upsilon (x) \,dx, (0 \le j_{1}\le \omega < j_{2}), \end{aligned}$$
(15)

respectively. Obviously, we have

$$\begin{aligned}{} & {} \mathfrak {J} ^{\alpha }_{(j_{1})^{+}}\Upsilon (\omega) = \big [\mathfrak {J} ^{\alpha }_{(j_{1})^{+}}\underline{\Upsilon } (\omega),\mathfrak {J}^{\alpha }_{(j_{1})^{+}}\overline{\Upsilon } (\omega)\big ], \end{aligned}$$
(16)

and

$$\begin{aligned}{} & {} \mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon (\omega) = \big [\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\underline{\Upsilon } (\omega), \mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\overline{\Upsilon } (\omega)\big ]. \end{aligned}$$
(17)

3 Main Results

In what follows, we obtain some results, which generalize Theorem 4, Theorem 5 and Theorem 6 of [42]. Particularly, all the functions considered in this section belong to \(IL_{I}\), all considered invex sets \(I\subseteq \mathbb {R}\) with respect to \(\xi :I \times I \rightarrow \mathbb {R}\) will mean \([j_{2}+\xi (j_{1},j_{2}),j_{2}]\), that is,

$$\begin{aligned} I=[j_{2}+\xi (j_{1},j_{2}),j_{2}]. \end{aligned}$$

Theorem 3.1

Let \(\Upsilon \in S(\hbar P, I, \mathbb {R}_{I}^{+})\), \(\xi\) satisfies Condition C, and \(\hbar (\frac{1}{2})>0\). Then,

$$\begin{aligned}&\frac{\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )}{\alpha \hbar (\frac{1}{2})} \supseteq \frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \left[ \mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}}\right. \left. \Upsilon (j_{2}) +\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\right] \supseteq \big [\Upsilon (j_{2})+\Upsilon (j_{2}+\xi (j_{1},j_{2}))\big ] \int _{0}^{1}t^{\alpha -1}\big [\hbar (t)+\hbar (1-t)\big ] \,dt \supseteq \big [(\hbar (0)+1)\Upsilon (j_{2})+\hbar (1)\Upsilon (j_{1})\big ] \int _{0}^{1}t^{\alpha -1}\big [\hbar (t)+\hbar (1-t)\big ] \,dt. \end{aligned}$$
(18)

Proof

By Condition C, we have

$$\begin{aligned}{} & {} \Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big ) =\Upsilon \bigg (j_{2}+(1-t)\xi (j_{1},j_{2})+\frac{1}{2}\xi \big (j_{2} +t\xi (j_{1},j_{2}),j_{2}+(1-t)\xi (j_{1},j_{2})\big )\bigg ). \end{aligned}$$
(19)

Since \(\Upsilon \in S(\hbar P, I, \mathbb {R}_{I}^{+})\), we get

$$\begin{aligned}{} & {} \Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )\supseteq \hbar (\frac{1}{2}) \bigg [\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) +\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\bigg ]. \end{aligned}$$
(20)

Multiplying by \(t^{\alpha -1}\) on both sides and integrating on [0, 1], we have

$$\begin{aligned} \begin{aligned}&\frac{\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )}{\hbar (\frac{1}{2})}\int _{0}^{1}t^{\alpha -1}\,dt \\&\supseteq \int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\,dt +\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\,dt. \end{aligned} \end{aligned}$$
(21)

That is,

$$\begin{aligned} \begin{aligned}&\frac{\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )}{\alpha \hbar (\frac{1}{2})} \supseteq \int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\,dt +\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\,dt. \end{aligned} \end{aligned}$$
(22)

To verify (18), let \(u=j_{2}+t\xi (j_{1},j_{2}), w=j_{2}+(1-t)\xi (j_{1},j_{2})\), then

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\,dt +\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\,dt \\&=\frac{(-1)^{\alpha -2 }}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \int _{j_{2} +\xi (j_{1},j_{2})}^{j_{2}}(j_{2}-u)^{\alpha -1}\Upsilon (u)\,du +\frac{(-1)^{\alpha }}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \int _{j_{2} +\xi (j_{1},j_{2})}^{j_{2}}\big (w-j_{2}-\xi (j_{1},j_{2})\big )^{\alpha -1}\Upsilon (w)\,dw =\frac{(-1)^{\alpha -2 }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}}\Upsilon (j_{2}) +\frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big ) =\frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \bigg [\mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}}\Upsilon (j_{2}) +\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )\bigg ]. \end{aligned} \end{aligned}$$
(23)

Consequently, we obtain

$$\begin{aligned} \begin{aligned}&\frac{\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )}{\alpha \hbar (\frac{1}{2})} \supseteq \frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \bigg [\mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}} \Upsilon (j_{2}) +\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\bigg ]. \end{aligned} \end{aligned}$$
(24)

On the other hand, by Condition C and the definition of \(\hbar\)-PIVF, we have

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) =\Upsilon \bigg (j_{2}+\xi (j_{1},j_{2})+t\xi \big (j_{2},j_{2} +\xi (j_{1},j_{2})\big )\bigg ) \supseteq \hbar (t)\Upsilon (j_{2}) +\hbar (1-t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big ), \end{aligned} \end{aligned}$$
(25)

and

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big ) =\Upsilon \bigg (j_{2}+\xi (j_{1},j_{2})+(1-t)\xi \big (j_{2},j_{2}+\xi (j_{1},j_{2})\big )\bigg ) \supseteq \hbar (t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar (1-t)\Upsilon (j_{2}). \end{aligned} \end{aligned}$$
(26)

Adding (25) and (26), multiplying by \(t^{\alpha -1}\) on both sides and integrating on [0, 1], then

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+(1-t) \xi (j_{1},j_{2})\big )\,dt+\int _{0}^{1}t^{\alpha -1} \Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\,dt \supseteq \int _{0}^{1}t^{\alpha -1}\big [\hbar (t)\Upsilon (j_{2})+\hbar (1-t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\big ]\,dt +\int _{0}^{1}t^{\alpha -1}\big [\hbar (t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar (1-t)\Upsilon (j_{2})\big ] \,dt =\big [\Upsilon (j_{2})+\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )\big ]\int _{0}^{1}t^{\alpha -1}\big [\hbar (t)+\hbar (1-t)\big ] \,dt \supseteq \big [(\hbar (0)+1)\Upsilon (j_{2}) +\hbar (1)\Upsilon (j_{1})\big ]\int _{0}^{1}t^{\alpha -1}\big [\hbar (t)+\hbar (1-t)\big ] \,dt. \end{aligned} \end{aligned}$$
(27)

Finally, (18) follows form (24) and (27). \(\square\)

Remark 3.2

Note that the proof of Theorem 3.1 in [43] is inaccurate. For example, the authors assume that \(x=j_{1}+\big (\frac{2-z}{2}\big )\xi (j_{2}, j_{1}), y=j_{1}+\frac{z}{2}\xi (j_{2}, j_{1})\). However, if we consider that \(\xi (j_{2},j_{1})=1-3\left|j_{2}-j_{1}\right|\) and \(j_{1}=1, j_{2}=2\), then \(x=z-1\in [-1, 1]\nsubseteq [0,2]\), \(y=1-z\in [-1, 1]\nsubseteq [0,2]\) and \(x+\frac{1}{2}\xi (y,x)=-\frac{7}{2}+4z \in [-\frac{7}{4}, \frac{9}{2}]\nsubseteq [0,2]\). Thus, some subsequent relevant proofs are incorrect. Next, we will give the correct forms of Theorem 3.1, Theorem 3.2 and Theorem 3.3 in [43] in the following remarks.

Remark 3.3

(1) If \(\hbar (t)=t\), then we obtain the correct form of Theorem 3.1 in [43].

(2) If \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then we get the Theorem 4.1 in [50].

(3) If \(\hbar (t)=t\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then Theorem 3.1 reduces to Theorem 1 in [40].

(4) If \(\hbar (t)=t^{s}\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then we can obtain Theorem 4 in [37].

(5) If \(\hbar (t)=1\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then we get the Remark 2 in [5].

(6) If \(\underline{\Upsilon }=\overline{\Upsilon }\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then we get the Theorem 6 in [41].

Example 3.4

Let \(\hbar (t)=t\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) for \(t\in [0,1]\), and \(\Upsilon (x)=[x^{2},10-e^{x}]\). If \(\alpha =2\), \(j_{1}=0\), \(j_{2}=1\), then we have

$$\begin{aligned}{} & {} \frac{\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )}{\alpha \hbar (\frac{1}{2})} =\bigg [\frac{1}{4},10-\sqrt{e} \bigg ], \frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \left[ \mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}}\Upsilon (j_{2})\right. +\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2} \left. +\xi (j_{1},j_{2})\big )\right] =\bigg [\frac{1}{3},11-e\bigg ],~ \big [\Upsilon (j_{2})+\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\big ] \int _{0}^{1}t^{\alpha -1}\big [\hbar (t)+\hbar (1-t)\big ] \,dt = \big [(\hbar (0)+1)\Upsilon (j_{2})+\hbar (1)\Upsilon (j_{1})\big ] \int _{0}^{1}t^{\alpha -1}\big [\hbar (t)+\hbar (1-t)\big ] \,dt =\bigg [\frac{1}{2}, \frac{19-e}{2}\bigg ]. \end{aligned}$$

As a result,

$$\begin{aligned}{} & {} \bigg [\frac{1}{4},10-\sqrt{e} \bigg ]\supseteq \bigg [\frac{1}{3},11-e\bigg ] \supseteq \bigg [\frac{1}{2}, \frac{19-e}{2}\bigg ]. \end{aligned}$$

Theorem 3.5

Let \(\Upsilon \in S(\hbar _{1} P, I, \mathbb {R}_{I}^{+})\), \(\varTheta \in S(\hbar _{2} P, I, \mathbb {R}_{I}^{+})\) and \(\xi\) satisfies Condition C. Then,

$$\begin{aligned} \begin{aligned}&\frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \left[ \mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}}\right. \Upsilon (j_{2})\varTheta (j_{2}) +\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2} \left. +\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )\right] \supseteq \varvec{U} (j_{1},j_{2})\int _{0}^{1}t^{\alpha -1} \big [\hbar _{1}(t)\hbar _{2}(t)+\hbar _{1}(1-t)\hbar _{2}(1-t)\big ] \,dt +\varvec{V} (j_{1},j_{2})\int _{0}^{1}t^{\alpha -1} \big [\hbar _{1}(t)\hbar _{2}(1-t)+\hbar _{1}(1-t)\hbar _{2}(t)\big ] \,dt, \end{aligned} \end{aligned}$$
(28)

where \(\varvec{U} (j_{1},j_{2})=\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )+\Upsilon (j_{2})\varTheta (j_{2})\), \(~ ~ ~ ~ ~ ~ ~ ~ ~\varvec{V} (j_{1},j_{2})=\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta (j_{2})+\Upsilon (j_{2})\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )\).

Proof

Since \(\Upsilon \in S(\hbar _{1} P, I, \mathbb {R}_{I}^{+})\) and \(\varTheta \in S(\hbar _{2} P, I, \mathbb {R}_{I}^{+})\), we have

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big ) \supseteq \hbar _{1}(t)\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )+\hbar _{1}(1-t)\Upsilon (j_{2}), \varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big ) \supseteq \hbar _{2}(t)\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big )+\hbar _{2}(1-t)\varTheta (j_{2}), \end{aligned} \end{aligned}$$
(29)

and

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) \supseteq \hbar _{1}(1-t)\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )+\hbar _{1}(t)\Upsilon (j_{2}),~\varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\supseteq \hbar _{2}(1-t)\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )+\hbar _{2}(t)\varTheta (j_{2}). \end{aligned} \end{aligned}$$
(30)

Further, we obtain

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big ) \supseteq \big [\hbar _{1}(t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{1}(1-t)\Upsilon (j_{2})\big ]\big [\hbar _{2}(t)\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )+\hbar _{2}(1-t)\varTheta (j_{2})\big ]=\hbar _{1}(t)\hbar _{2}(t)\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{1}(t)\hbar _{2}(1-t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta (j_{2}) +\hbar _{1}(1-t)\hbar _{2}(t)\Upsilon (j_{2})\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{1}(1-t)\hbar _{2}(1-t)\Upsilon (j_{2})\varTheta (j_{2}). \end{aligned} \end{aligned}$$
(31)

Similarly,

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\supseteq \big [\hbar _{1}(1-t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{1}(t)\Upsilon (j_{2})\big ]\big [\hbar _{2}(1-t)\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{2}(t)\varTheta (j_{2})\big ]\qquad =\hbar _{1}(1-t)\hbar _{2}(1-t)\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{1}(1-t)\hbar _{2}(t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta (j_{2})\qquad +\hbar _{1}(t)\hbar _{2}(1-t)\Upsilon (j_{2})\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big ) +\hbar _{1}(t) \hbar _{2}(t)\Upsilon (j_{2})\varTheta (j_{2}). \end{aligned} \end{aligned}$$
(32)

Consequently, we have

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\varTheta \big (j_{2} +t\xi (j_{1},j_{2})\big )+\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) \supseteq \hbar _{1}(t)\hbar _{2}(t)\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{1}(t)\hbar _{2}(1-t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta (j_{2}) +\hbar _{1}(1-t)\hbar _{2}(t)\Upsilon (j_{2})\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big ) +\hbar _{1}(1-t)\hbar _{2}(1-t)\Upsilon (j_{2})\varTheta (j_{2}) +\hbar _{1}(1-t)\hbar _{2}(1-t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big )+\hbar _{1}(t) \hbar _{2}(t)\Upsilon (j_{2})\varTheta (j_{2}) +\hbar _{1}(1-t)\hbar _{2}(t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta (j_{2}) +\hbar _{1}(t) \hbar _{2}(1-t)\Upsilon (j_{2})\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big ) =\big [\hbar _{1}(t)\hbar _{2}(t)+\hbar _{1}(1-t)\hbar _{2}(1-t)\big ] \big [\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big )+\Upsilon (j_{2})\varTheta (j_{2})\big ] +\big [\hbar _{1}(t)\hbar _{2}(1-t)+\hbar _{1}(1-t) \hbar _{2}(t)\big ]\big [\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )\varTheta (j_{2})+\Upsilon (j_{2})\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big )\big ]. \end{aligned} \end{aligned}$$
(33)

Multiplying by \(t^{\alpha -1}\) on both sides and integrating on [0, 1]. To obtain (28), let

$$\begin{aligned} u=j_{2}+t\xi (j_{1},j_{2}),w=j_{2}+(1-t)\xi (j_{1},j_{2}). \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\,dt +\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big )\,dt =\frac{(-1)^{\alpha }}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \int _{j_{2} +\xi (j_{1},j_{2})}^{j_{2}}\big (w-j_{2}-\xi (j_{1},j_{2})\big )^{\alpha -1}\Upsilon (w)\varTheta (w)\,dw +\frac{(-1)^{\alpha -2 }}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \int _{j_{2}+\xi (j_{1},j_{2})}^{j_{2}}(j_{2}-u)^{\alpha -1}\Upsilon (u)\varTheta (u)\,du =\frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \left[ \mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}}\Upsilon (j_{2})\varTheta (j_{2}) \right. +\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta \big (j_{2} \left. +\xi (j_{1},j_{2})\big )\right] . \end{aligned} \end{aligned}$$
(34)

According to (33), we have

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\,dt +\int _{0}^{1}t^{\alpha -1}\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big )\,dt\supseteq \int _{0}^{1}t^{\alpha -1}\big [\hbar _{1}(t)\hbar _{2}(t) +\hbar _{1}(1-t)\hbar _{2}(1-t)\big ]\varvec{U} (j_{1},j_{2})\,dt +\int _{0}^{1}t^{\alpha -1}\big [\hbar _{1}(t)\hbar _{2}(1-t) +\hbar _{1}(1-t) \hbar _{2}(t)\big ]\varvec{V} (j_{1},j_{2})\,dt =\varvec{U} (j_{1},j_{2})\int _{0}^{1}t^{\alpha -1} \big [\hbar _{1}(t)\hbar _{2}(t)+\hbar _{1}(1-t)\hbar _{2}(1-t)\big ] \,dt +\varvec{V} (j_{1},j_{2})\int _{0}^{1}t^{\alpha -1} \big [\hbar _{1}(t)\hbar _{2}(1-t)+\hbar _{1}(1-t)\hbar _{2}(t)\big ] \,dt. \end{aligned} \end{aligned}$$
(35)

From the above inequalities (34) and (35), we obtain (28). \(\square\)

Remark 3.6

(1) If \(\hbar _{1}(t)=\hbar _{2}(t)=t\), then we obtain the correct form of Theorem 3.2 in [43].

(2) If \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then Theorem 3.5 reduces to Theorem 4.5 in [50].

(3) If \(\underline{\Upsilon }=\overline{\Upsilon }\), \(\underline{\varTheta }=\overline{\varTheta }\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then we get Theorem 7 in [41].

(4) If \(\underline{\Upsilon }=\overline{\Upsilon }\), \(\underline{\varTheta }=\overline{\varTheta }\), \(\hbar _{1}(t)=\hbar _{2}(t)=t\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then we get Theorem 1 in [38].

(5) If \(\hbar _{1}(t)=\hbar _{2}(t)=t\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then Theorem 3.5 reduces to the result for convex IVFs:

$$\begin{aligned} \begin{aligned}&\frac{1}{ j_{2}-j_{1}}\int _{j_{1}}^{j_{2}}\Upsilon (x)\varTheta (x)\,dx \supseteq \frac{1}{3}\big [\Upsilon (j_{2})\varTheta (j_{2})+\Upsilon (j_{1})\varTheta (j_{1})\big ] +\frac{1}{6}\big [\Upsilon (j_{1})\varTheta (j_{2})+\Upsilon (j_{2})\varTheta (j_{1})\big ]. \end{aligned} \end{aligned}$$
(36)

(6) If \(\hbar _{1}(t)=\hbar _{2}(t)=t^{s}\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then Theorem 3.5 reduces to the result for s-convex IVFs:

$$\begin{aligned} \begin{aligned}&\frac{1}{ j_{2}-j_{1}}\int _{j_{1}}^{j_{2}}\Upsilon (x)\varTheta (x) dx \supseteq \frac{\big [\Upsilon (j_{2})\varTheta (j_{2})+\Upsilon (j_{1})\varTheta (j_{1})\big ] }{2s+1} +\big [\Upsilon (j_{1})\varTheta (j_{2})+\Upsilon (j_{2})\varTheta (j_{1})\big ] \int _{0}^{1} \big (t(1-t)\big )^{s}\,dt. \end{aligned} \end{aligned}$$
(37)

Theorem 3.7

Let \(\Upsilon \in S(\hbar _{1} P, I, \mathbb {R}_{I}^{+})\), \(\varTheta \in S(\hbar _{2} P, I, \mathbb {R}_{I}^{+})\), \(\xi\) satisfies Condition C, \(\hbar _{1}(\frac{1}{2})>0\), and \(\hbar _{2}(\frac{1}{2})>0\). Then,

$$\begin{aligned} \begin{aligned}&\frac{\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )}{\alpha \hbar _{1}(\frac{1}{2})\hbar _{2}(\frac{1}{2})} \supseteq \frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \bigg [\mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}}\Upsilon (j_{2})\varTheta (j_{2}) +\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta (j_{2}+\xi (j_{1},j_{2}))\bigg ] +\varvec{V} (j_{1},j_{2})\int _{0}^{1}t^{\alpha -1} \big [\hbar _{1}(t)\hbar _{2}(t)+\hbar _{1}(1-t)\hbar _{2}(1-t)\big ] \,dt +\varvec{U}(j_{1},j_{2})\int _{0}^{1}t^{\alpha -1} \big [\hbar _{1}(t)\hbar _{2}(1-t)+\hbar _{1}(1-t)\hbar _{2}(t)\big ] \,dt, \end{aligned} \end{aligned}$$
(38)

where \(\varvec{U} (j_{1},j_{2})=\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )+\Upsilon (j_{2})\varTheta (j_{2})\), \(~ ~ ~ ~ ~ ~ ~ ~ ~\varvec{V} (j_{1},j_{2})=\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta (j_{2})+\Upsilon (j_{2})\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )\).

Proof

By Condition C, we have

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big ) =\Upsilon \bigg (j_{2}+(1-t)\xi (j_{1},j_{2}) +\frac{1}{2}\xi \big (j_{2}+t\xi (j_{1},j_{2}),j_{2}+(1-t)\xi (j_{1},j_{2})\big )\bigg ) \cdot \varTheta \bigg (j_{2}+(1-t)\xi (j_{1},j_{2}) +\frac{1}{2}\xi \big (j_{2}+t\xi (j_{1},j_{2}),j_{2}+(1-t)\xi (j_{1},j_{2})\big )\bigg ).\qquad \end{aligned} \end{aligned}$$
(39)

Since \(\Upsilon \in S(\hbar _{1} P, I, \mathbb {R}_{I}^{+})\) and \(\varTheta \in S(\hbar _{2} P, I, \mathbb {R}_{I}^{+})\), we obtain

$$\begin{aligned} \begin{aligned}&\frac{\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )}{\hbar _{1}(\frac{1}{2})\hbar _{2}(\frac{1}{2})} \supseteq \big [\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) +\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\big ]\big [\varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) +\varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big )\big ] =\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) +\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big ) +\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) +\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big ). \end{aligned} \end{aligned}$$
(40)

To prove (38), multiplying by \(t^{\alpha -1}\) on both sides of (40) and integrating on [0, 1]. By (34),

$$\begin{aligned} \begin{aligned}&\frac{\Upsilon \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+\frac{1}{2}\xi (j_{1},j_{2})\big )}{\alpha \hbar _{1}(\frac{1}{2})\hbar _{2}(\frac{1}{2})} \supseteq \frac{(-1)^{\alpha }\Gamma (\alpha )}{\big (\xi (j_{1},j_{2})\big )^{\alpha }} \bigg [\mathfrak {J} ^{\alpha }_{\big (j_{2}+\xi (j_{1},j_{2})\big )^{+}}\Upsilon (j_{2})\varTheta (j_{2}) +\mathfrak {J} ^{\alpha }_{(j_{2})^{-}}\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta (j_{2}+\xi (j_{1},j_{2}))\bigg ] +\int _{0}^{1} t^{\alpha -1} \Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big )\,dt +\int _{0}^{1}t^{\alpha -1} \Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\,dt. \end{aligned} \end{aligned}$$
(41)

In addition,

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big ) \supseteq \big [\hbar _{1}(1-t)\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )+\hbar _{1}(t)\Upsilon (j_{2})\big ]\big [\hbar _{2}(t)\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )+\hbar _{2}(1-t)\varTheta (j_{2})\big ] =\hbar _{1}(1-t)\hbar _{2}(t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big )+\hbar _{1}(1-t)\hbar _{2}(1-t)\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )\varTheta (j_{2}) +\hbar _{1}(t)\hbar _{2}(t)\Upsilon (j_{2})\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{1}(t) \hbar _{2}(1-t)\Upsilon (j_{2})\varTheta (j_{2}). \end{aligned} \end{aligned}$$
(42)

Similarly,

$$\begin{aligned} \begin{aligned}&\Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big )\varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\supseteq \big [\hbar _{1}(t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big ) +\hbar _{1}(1-t)\Upsilon (j_{2})\big ]\big [\hbar _{2}(1-t)\varTheta \big (j_{2}+\xi (j_{1},j_{2})\big )+\hbar _{2}(t)\varTheta (j_{2})\big ] =\hbar _{1}(t)\hbar _{2}(1-t)\Upsilon \big (j_{2}+\xi (j_{1},j_{2})\big )\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big )+\hbar _{1}(t)\hbar _{2}(t)\Upsilon \big (j_{2} +\xi (j_{1},j_{2})\big )\varTheta (j_{2}) +\hbar _{1}(1-t)\hbar _{2}(1-t)\Upsilon (j_{2})\varTheta \big (j_{2} +\xi (j_{1},j_{2})\big ) +\hbar _{1}(1-t) \hbar _{2}(t)\Upsilon (j_{2})\varTheta (j_{2}). \end{aligned} \end{aligned}$$
(43)

Combining (42) and (43), multiplying by \(t^{\alpha -1}\) on both sides and integrating on [0, 1], we have

$$\begin{aligned} \begin{aligned}&\int _{0}^{1} t^{\alpha -1} \Upsilon \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+t\xi (j_{1},j_{2})\big )\,dt +\int _{0}^{1} t^{\alpha -1} \Upsilon \big (j_{2}+t\xi (j_{1},j_{2})\big ) \varTheta \big (j_{2}+(1-t)\xi (j_{1},j_{2})\big )\,dt \supseteq \int _{0}^{1}t^{\alpha -1}\big [\hbar _{1}(t)\hbar _{2}(t) +\hbar _{1}(1-t)\hbar _{2}(1-t)\big ]\varvec{V} (j_{1},j_{2})\,dt +\int _{0}^{1}t^{\alpha -1}\big [\hbar _{1}(t)\hbar _{2}(1-t)+\hbar _{1}(1-t) \hbar _{2}(t)\big ]\varvec{U}(j_{1},j_{2})\,dt =\varvec{V} (j_{1},j_{2})\int _{0}^{1}t^{\alpha -1} \big [\hbar _{1}(t)\hbar _{2}(t)+\hbar _{1}(1-t)\hbar _{2}(1-t)\big ] \,dt +\varvec{U}(j_{1},j_{2})\int _{0}^{1}t^{\alpha -1} \big [\hbar _{1}(t)\hbar _{2}(1-t)+\hbar _{1}(1-t)\hbar _{2}(t)\big ] \,dt. \end{aligned} \end{aligned}$$
(44)

From the above inequalities (41) and (44), we obtain (38). \(\square\)

Remark 3.8

(1) If \(\hbar _{1}(t)=\hbar _{2}(t)=t\), then we obtain the correct form of Theorem 3.3 in [43].

(2) If \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then Theorem 3.7 reduces to Theorem 4.6 in [50].

(3) If \(\underline{\Upsilon }=\overline{\Upsilon }\), \(\underline{\varTheta }=\overline{\varTheta }\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), we get Theorem 8 in [41].)

(4) If \(\underline{\Upsilon }=\overline{\Upsilon }\), \(\underline{\varTheta }=\overline{\varTheta }\), \(\hbar _{1}(t)=\hbar _{2}(t)=t\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then we get Theorem 1 in [38].

(5) If \(\hbar _{1}(t)=\hbar _{2}(t)=t\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then Theorem 3.7 reduces to the result for convex IVFs:

$$\begin{aligned} \begin{aligned}&2\Upsilon \big (\frac{j_{1}+j_{2}}{2}\big ) \varTheta \big (\frac{j_{1}+j_{2}}{2}\big ) \supseteq \frac{1}{\big ( j_{2}-j_{1}\big )} \int _{j_{1}}^{j_{2}}\Upsilon (x)\varTheta (x)\,dx +\frac{1}{3}\big [\Upsilon (j_{1})\varTheta (j_{2})+\Upsilon (j_{2})\varTheta (j_{1})\big ] +\frac{1}{6}\big [\Upsilon (j_{2})\varTheta (j_{2})+\Upsilon (j_{1})\varTheta (j_{1})\big ]. \end{aligned} \end{aligned}$$
(45)

(6) If \(\hbar _{1}(t)=\hbar _{2}(t)=t^{s}\), \(\xi (j_{1},j_{2})=j_{1}-j_{2}\) and \(\alpha =1\), then Theorem 3.7 reduces to the result for s-convex IVFs:

$$\begin{aligned} \begin{aligned}&2^{2s-1}\Upsilon \big (\frac{j_{1}+j_{2}}{2}\big ) \varTheta \big (\frac{j_{1}+j_{2}}{2}\big ) \supseteq \frac{1}{\big ( j_{2}-j_{1}\big )} \int _{j_{1}}^{j_{2}}\Upsilon (x)\varTheta (x)\,dx +\frac{\big [\Upsilon (j_{1})\varTheta (j_{2})+\Upsilon (j_{2})\varTheta (j_{1})\big ] }{2s+1} +\big [\Upsilon (j_{2})\varTheta (j_{2})+\Upsilon (j_{1})\varTheta (j_{1})\big ] \int _{0}^{1}\big (t(1-t)\big )^{s}\,dt. \end{aligned} \end{aligned}$$
(46)

4 Conclusion

The topic of H–H-type inequalities for IVFs and their extensions has gained significant attention in the literature due to its wide range of applications. In this paper, we contribute to this field by establishing new H–H and Pachpatte-type inequalities for \(\hbar\)-PIVFs. Theorems 3.1, 3.5, and 3.7 presented in our paper provide novel results that not only improve upon the main theorems proposed by Srivastava et al., but also generalize the conclusions found in the existing literature. These new inequalities offer enhanced insights into the properties and behavior of IVFs, particularly those exhibiting \(\hbar\)-preinvex.

By introducing these new results, we aim to inspire further investigations in this area. We believe that our findings will encourage researchers to explore more general inequalities and their applications, and our work serves as a stepping stone for others to delve deeper into the realm of IVFs and inequalities. In our future research, we plan to extend our study to encompass H–H, Fejér, Jensen, and Pachpatte-type inequalities for \(\hbar\)-PIVFs, as well as for fuzzy IVFs. In addition, we intend to explore these inequalities within the context of generalized fractional integral, post-quantum calculus, and quantum integral. These investigations will have broad implications in various domains, including artificial intelligence, optimization engineering, financial activities, and so forth.