Abstract
We present a comprehensive study on Hermite–Hadamard-type inequalities for interval-valued functions that are \(\hbar\)-preinvex, using the Riemann–Liouville fractional integral. Our research extends and generalizes some existing results found in the literature. In addition, we provide accurate proofs for the main theorems originally derived by Srivastava et al. in their publication titled “Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators" (Int. J. Comput. Int. Sys. 15(1):8, 2022). Finally, we illustrate our findings through a practical example to demonstrate the validity of our results.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
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
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
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
For \(\eta \in \mathbb {R}\) and \(\alpha \in \mathbb {R}_{I}\), \(\eta \alpha\) is defined by
For arbitrary \(\alpha , \beta \in \mathbb {R}_{I}\), the four arithmetic operators \((+, -, \cdot , / )\) are defined by
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
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
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
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 \(t\in [0,1]\),
and
From Condition C, we also have
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
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
and
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
and
respectively. Obviously, we have
and
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,
Theorem 3.1
Let \(\Upsilon \in S(\hbar P, I, \mathbb {R}_{I}^{+})\), \(\xi\) satisfies Condition C, and \(\hbar (\frac{1}{2})>0\). Then,
Proof
By Condition C, we have
Since \(\Upsilon \in S(\hbar P, I, \mathbb {R}_{I}^{+})\), we get
Multiplying by \(t^{\alpha -1}\) on both sides and integrating on [0, 1], we have
That is,
To verify (18), let \(u=j_{2}+t\xi (j_{1},j_{2}), w=j_{2}+(1-t)\xi (j_{1},j_{2})\), then
Consequently, we obtain
On the other hand, by Condition C and the definition of \(\hbar\)-PIVF, we have
and
Adding (25) and (26), multiplying by \(t^{\alpha -1}\) on both sides and integrating on [0, 1], then
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
As a result,
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,
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
and
Further, we obtain
Similarly,
Consequently, we have
Multiplying by \(t^{\alpha -1}\) on both sides and integrating on [0, 1]. To obtain (28), let
Then,
According to (33), we have
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:
(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:
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,
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
Since \(\Upsilon \in S(\hbar _{1} P, I, \mathbb {R}_{I}^{+})\) and \(\varTheta \in S(\hbar _{2} P, I, \mathbb {R}_{I}^{+})\), we obtain
To prove (38), multiplying by \(t^{\alpha -1}\) on both sides of (40) and integrating on [0, 1]. By (34),
In addition,
Similarly,
Combining (42) and (43), multiplying by \(t^{\alpha -1}\) on both sides and integrating on [0, 1], we have
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:
(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:
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.
Availability of Data and Material
Not applicable.
References
Arqub, O.A.: Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations. Neural Comput. Appl. 28, 1591–1610 (2017)
Arqub, O.A., Singh, J., Alhodaly, M.: Adaptation of kernel functions-based approach with Atangana-Baleanu-Caputo distributed order derivative for solutions of fuzzy fractional Volterra and Fredholm integrodifferential equations. Math. Methods Appl. Sci. 46(7), 7807–7834 (2023)
Arqub, O.A., Singh, J., Maayah, B., Alhodaly, M.: Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag-Leffler kernel differential operator. Math. Methods Appl. Sci. 46(7), 7965–7986 (2023)
Ahmad, I., Jayswal, A., Banerjee, J.: On interval-valued optimization problems with generalized invex functions. J. Inequal. Appl. 14, 313 (2013)
An, Y.R., Ye, G.J., Zhao, D.F., Liu, W.: Hermite-Hadamard type inequalities for interval (\(h_{1}, h_{2}\))-convex functions. Mathematics 7(5), 436 (2019)
Awan, M.U., Akhtar, N., Iftikhar, S., Noor, M.A., Chu, Y.-M.: New Hermite-Hadamard type inequalities for n-polynomial harmonically convex functions. J. Inequal. Appl. 2020(1), 1–12 (2020)
Budak, H., Kashuri, A., Butt, S.I.: Fractional Ostrowski type inequalities for interval valued functions. Filomat 36(8), 2531–2540 (2022)
Budak, H., Tunç, T., Sarikaya, M.: Fractional Hermite-Hadamard-type inequalities for interval-valued functions. Proc. Am. Math. Soc. 148(2), 705–718 (2020)
Burkill, J.C.: Functions of Intervals. Proc. Lond. Math. Soc. 2(22), 275–310 (1924)
Chalco-Cano, Y., Flores-Franulič, A., Román-Flores, H.: Ostrowski type inequalities for interval-valued functions using generalized Hukuhara derivative. Comput. Appl. Math. 31, 457–472 (2012)
Chen, S.-M., Liao, W.-T.: Multiple attribute decision making using Beta distribution of intervals, expected values of intervals, and new score function of interval-valued intuitionistic fuzzy values. Inform. Sci. 579, 863–887 (2021)
Chen, S.-M., Yu, S.-H.: Multiattribute decision making based on novel score function and the power operator of interval-valued intuitionistic fuzzy values. Inform. Sci. 606, 763–785 (2022)
Costa, T.M., Chalco-Cano, Y., Román-Flores, H.: Wirtinger-type integral inequalities for interval-valued functions. Fuzzy Sets Syst. 396, 102–114 (2020)
Costa, T.M., Román-Flores, H., Chalco-Cano, Y.: Opial-type inequalities for interval-valued functions. Fuzzy Sets Syst. 358, 48–63 (2019)
Du, T.S., Zhou, T.C.: On the fractional double integral inclusion relations having exponential kernels via interval-valued co-ordinated convex mappings. Chaos Soliton Fractals 156, 111846 (2022)
Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)
Guo, Y.T., Ye, G.J., Liu, W., Zhao, D.F., Treanţă, S.: On symmetric gH-derivative: applications to dual interval-valued optimization problems. Chaos Soliton Fractals 158, 112068 (2022)
İşcan, İ: Hermite-Hadamard type inequalities for harmonically convex functions. Hacet. J. Math. Stat. 43(6), 935–942 (2014)
Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis. Springer-Verlag, London (2001)
Khan, M.B., Noor, M.A., Abdeljawad, T., Mousa, A.A.A., Abdalla, B., Alghamdi, S.M.: LR-preinvex interval-valued functions and Riemann-Liouville fractional integral inequalities. Fractal Fract. 5(4), 243 (2021)
Khan, M.B., Noor, M.A., Mohammed, P.O., Guirao, J.L.G., Noor, K.I.: Some integral inequalities for generalized convex fuzzy-interval-valued functions via fuzzy Riemann integrals. Int. J. Comput. Int. Sys. 14(1), 158 (2021)
Khan, M.B., Srivastava, H.M., Mohammed, P.O., Macías-Díaz, J.E., Hamed, Y.: Some new versions of integral inequalities for log-preinvex fuzzy-interval-valued functions through fuzzy order relation. Alex. Eng. J. 61(9), 7089–7101 (2022)
Kolmogorov, A.N.: Untersuchungen über integralbegriff. Math. Ann 103, 654–696 (1930)
Latif, M.A., Dragomir, S.S., Momoniat, E.: Some weighted integral inequalities for differentiable h-preinvex functions. Georgian Math. J. 25(3), 441–450 (2018)
Latif, M.A., Kashuri, A., Hussain, S., Delavar, R.M.: Trapezium-type inequalities for h-preinvex functions and their applications. Filomat 36(10), 3393–3404 (2022)
Li, D.-F.: Models and Methods for Interval-Valued Cooperative Games in Economic management. Springer, Cham (2016)
Lupulescu, V.: Fractional calculus for interval-valued functions. Fuzzy Sets Syst. 265, 63–85 (2015)
Matłoka, M.: Inequalities for h-preinvex functions. Appl. Math. Comput. 234, 52–57 (2014)
Matłoka, M.: Hermite-Hadamard type inequalities for h-preinvex mappings via fractional integrals. Control Cybernet. 44(2), 275–285 (2015)
Matłoka, M.: Relative h-preinvex functions and integral inequalities. Georgian Math. J. 27(2), 285–295 (2020)
Mohan, S.R., Neogy, S.K.: On invex sets and preinvex Functions. J. Math. Anal. Appl. 189(3), 901–908 (1995)
Moore, R.E.: Interval Analysis. Prentice-Hall Inc, Englewood Cliffs (1966)
Moore, R.E.: Methods and Applications of Interval Analysis. SIAM, Philadelphia (1979)
Noor, M.A.: Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory. 2(2), 126–131 (2007)
Noor, M.A., Noor, K.I., Awan, M.U.: A new Hermite-Hadamard type inequality for h-convex functions. Creat. Math. Inform. 24(2), 191–197 (2015)
Noor, M.A., Noor, K.I., Awan, M.U., Li, J.: On Hermite-Hadamard inequalities for h-preinvex functions. Filomat 28(7), 1463–1474 (2014)
Osuna-Gómez, R., Jiménez-Gamero, M.D., Chalco-Cano, Y., Rojas-Medar, M.A.: Hadamard and Jensen inequalities for s-convex fuzzy processes. In: Soft Methodology and Random Information Systems, pp. 645–652. Springer, Berlin (2004)
Pachpatte, B.G.: On some inequalities for convex functions. RGMIA. Res. Rep. Coll. 6(1), 1–9 (2003)
Podlubny, I.: Fractional Differential Equations, Mathematics in Science and Engineering. Academic Press Inc, San Diego (1999)
Sadowska, E.: Hadamard inequality and a refinement of Jensen inequality for set-valued functions. Result. Math. 32, 332–337 (1997)
Sarikaya, M.Z., Saglam, A., Yildirim, H.: On some Hadamard-type inequalities for h-convex functions. J. Math. Inequal. 2(3), 335–341 (2008)
Sharma, N., Singh, S.K., Mishra, S.K., Hamdi, A.: Hermite-Hadamard-type inequalities for interval-valued preinvex functions via Riemann-Liouville fractional integrals. J. Inequal. Appl. 1–15, 2021 (2021)
Srivastava, H.M., Sahoo, S.K., Mohammed, P.O., Baleanu, D., Kodamasingh, B.: Hermite-Hadamard type inequalities for interval-valued preinvex functions via fractional integral operators. Int. J. Comput. Int. Sys. 15(1), 8 (2022)
Srivastava, H.M., Sahoo, S.K., Mohammed, P.O., Kodamasingh, B., Hamed, Y.S.: New Riemann-Liouville fractional-order inclusions for convex functions via interval-valued settings associated with pseudo-order relations. Fractal Fract. 6(4), 212 (2022)
Sun, W.: Some Hermite-Hadamard type inequalities for generalized h-preinvex function via local fractional integrals and their applications. Adv. Differ. Equ. 2020(1), 1–14 (2020)
Sun, Y.Y., Zhang, X.Y., Wan, A.T.K., Wang, S.Y.: Model averaging for interval-valued data. Eur. J. Oper. Res. 301(2), 772–784 (2022)
Weir, T., Mond, B.: Pre-invex functions in multiple objective optimization. J. Math. Anal. Appl. 136(1), 29–38 (1988)
Wu, H.-C.: Solving the interval-valued optimization problems based on the concept of null set. J. Ind. Manag. Optim. 14(3), 1157–1178 (2018)
Zhao, D.F., Ali, M.A., Kashuri, A., Budak, H., Sarikaya, M.Z.: Hermite-Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals. J. Inequal. Appl. 2020(1), 1–38 (2020)
Zhao, D.F., An, T.Q., Ye, G.J., Liu, W.: New Jensen and Hermite-Hadamard type inequalities for h-convex interval-valued functions. J. Inequal. Appl. 1–14, 2018 (2018)
Zhao, D.F., An, T.Q., Ye, G.J., Liu, W.: Chebyshev type inequalities for interval-valued functions. Fuzzy Sets Syst. 396, 82–101 (2020)
Zhao, D.F., An, T.Q., Ye, G.J., Liu, W.: Some generalizations of Opial type inequalities for interval-valued functions. Fuzzy Sets Syst. 436, 128–151 (2022)
Zhao, D.F., An, T.Q., Ye, G.J., Torres, D.F.M.: On Hermite-Hadamard type inequalities for harmonical \(h\)-convex interval-valued functions. Math. Inequal. Appl. 23(1), 95–105 (2020)
Acknowledgements
The authors are very grateful to two anonymous referees, for several valuable and helpful comments, suggestions and questions, which helped to improve the paper into its present form.
Funding
The work was supported by the Open Fund of National Cryosphere Desert Data Center of China (2021kf03) and Foundation of Hubei Normal University (2022055).
Author information
Authors and Affiliations
Contributions
All the authors contributed equally to the writing of this manuscript. They also read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Tan, Y., Zhao, D. & Sarikaya, M.Z. Hermite–Hadamard-type Inequalities for \(\hbar\)-preinvex Interval-Valued Functions via Fractional Integral. Int J Comput Intell Syst 16, 120 (2023). https://doi.org/10.1007/s44196-023-00300-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s44196-023-00300-y