Hermite–Hadamard and Fractional Integral Inequalities for Interval-Valued Generalized <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>-</span>Convex Function

Li, Zhengbo; Kamran, undefined; Zahoor, Muhammad Sajid; Akhtar, Huma

doi:https://doi.org/10.1155/2020/4606439

Journal of Mathematics

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 4606439 | https://doi.org/10.1155/2020/4606439

Hermite–Hadamard and Fractional Integral Inequalities for Interval-Valued Generalized -Convex Function

Zhengbo Li,¹ Kamran,²Muhammad Sajid Zahoor,³and Huma Akhtar³

Academic Editor: Sunil Kumar

Received23 Aug 2020

Revised07 Sept 2020

Accepted25 Sept 2020

Published21 Oct 2020

Abstract

In the present paper, the new interval-valued generalized convex functions are introduced. By using the notion of interval-valued generalized convex functions and some auxiliary results of interval analysis, new Hermite–Hadamard and Fejér type inequalities are proved. The established results are more generalized than existing results in the literature. Moreover, fractional integral inequality for this generalization is also established.

1. Introduction

The theory of interval analysis introduced in numerical analysis by Moore in [1] had rapid development in last few decades. In computational problems, one can program a computer to find interval that contains the exact answer to the problems. Also, interval analysis provides rigorous enclosure of solution to the model equation. Moreover, the interval analysis is widely used in chemical and structured engineering, economics, control circuitry design, robotics, beam physics, behavioral ecology, constraint satisfaction, computer graphics, signal processing, asteroid orbits and global optimization [2], neural network output optimization [3], and many others. For interesting fundamental results, we refer [2, 4–8] to the readers.

Since the convexity play a vital role not only in convex analysis but also in almost all branches of mathematics. The famous inequalities in convex analysis are Jensen type, Hermite–Hadamard type, Fejér type, Ostrowski type, etc. For deeper insight about these inequalities, we refer [9–16] and references therein.

Furthermore, the definition of classical convexity enables us to tackle modern applied problems, because most of the problems are nonconvex in nature. Famous generalization of convexity are logarithmic convexity [16], -convexity [17], convexity [18], -convexity [19], modified -convexity [15], etc. For example, in [20], Nchama et al. used the CaputoFabrizio fractional integral and gave some new inequalities. For detailed applications of fractional calculus, we refer [21–28] to the readers and references therein.

In order to introduce the main definition of this paper, let us recall few generalizations of convexity present in the literature.

Definition 1 (see [17]). An interval is -convex set, if for any , , we havewhere or , , , and .

Definition 2 (see [17]). A mapping defined from a -convex set to is said to be -convex function, iffor each and hold.

Definition 3 (see [29]). The mapping defined from to is said to be -convex ifholds with respect to for appropriate , and for each , .

Definition 4 (see [29]). A mapping is nonnegatively homogeneous if for each and .

Definition 5 (see [30]). A mapping defined from a -convex set to is said to be generalized convex function, ifholds for be a bifunction for appropriate and for each and .
Now, we present the concept of interval-valued generalized convex function.

Definition 6. A mapping defined from a -convex set to is said to be interval-valued generalized -convex function, ifholds for be a bifunction for appropriate and for each and .
Here, for and = 1, (5) is an -convexity, for and (5) is -convexity, and for = 1 and , (5) is classical convexity.
This article is in the direction of the concepts and some results discussed in [30], but now we use interval-valued generalized -convex function instead of generalized convex function. After this introduction, in Section 2, we develop some basic properties of interval-valued generalized convex functions. InSection 3, we make some new inequalities like Hermite–Hadamard’s and Fejér type for interval-valued generalized convex functions.

2. Basic Results

Here, we derive some operations which preserves interval-valued generalized convex function.

Proposition 1. Let and be two interval-valued generalized convex functions:(1)If is additive, then is interval-valued generalized convex(2)If is nonnegatively homogeneous, then is interval-valued generalized convex for any .

Proof. The proof is straightforward.

Theorem 1. Let be an interval-valued function such that , then iff and .

Proof. Let , then for any , we havethat is,It follows thatThis shows thatConversely, suppose thatThen, it follows that . This completes the proof.

Theorem 2. Let be an interval-valued function such that , then if and .

Proof. The proof is similar to that of Theorem 1.

3. Hermite–Hadamard-Type Inequality for Interval-Valued Generalized Convex Function

In the following theorem, we present the Hermite–Hadamard type inequality for interval-valued generalized convex function.

Theorem 3. Let be an interval-valued generalized convex function for with condition , then we obtain the following inequality:

Proof. Take and , it impliesSo,By definition of interval-valued generalized convex functions, we haveNow, by the definition of intervalwe haveIt follows thatIntegrating (17) with respect to “” on [0, 1], we getwhich impliesNow,Similarly,Adding (21) and (22), we obtainNow, Integrating (18) with respect to “” on [0, 1], we getwhich impliesNow,Similarly,Adding (26) and (27), we obtainCombining (20) and (21), we obtainCombining (25) and (28), we obtainEquations (29) and (30) follows:which completely follows (11)

Remark 1. By putting and , (11) becomes Hermite–Hadamard type inequality for -convexity [18].

Remark 2. By putting and in (11), we obtain Hermite–Hadamard type inequality for -convexity [17].

Remark 3. By putting , = 1 and in (11), we get classical Hermite–Hadamard type inequality for convex functions.

Example 1. Consider and be defined by with as an odd number, then we havePut and simplify, we getCombining (32), (34), and (35), we get

4. Fejér-Type Inequality for Interval-Valued Generalized Convex Function

Now, we develop Fejér type inequality for interval-valued generalized convex functions.

Theorem 4. Let be nonnegative interval-valued generalized convex functions such that , thenwhere

Proof. Since and are interval-valued generalized convex functions, we havefor all . Since and are nonnegative,By the definition of interval, we haveIt followsIntegrating (42) over , we obtain the following inequality:Setting , we getIntegrating (43) over (0, 1), we getSetting , we getUsing inequality, (47) we getThen, we obtain the inequality (37).

Remark 4. If we put , and in (37), then it reduces to classical convex functions.

5. Fractional Hermite–Hadamard-Type Inequalities for Interval-Valued Generalized Convex Functions

The fractional inequalities has applications in every field of science and engineering. The new fractional integral inequalities in analysis are always appreciable. Because of the wide applications of Hermite–Hadamard-type inequalities and fractional integrals, many researchers extended their studies to Hermite–Hadamard-type inequality involving fractional integral inequalities. For fractional integral inequalities for interval-valued function, we suggest the reader to refer [31, 32].

Definition 7 (see [33–35]). Let . The right-hand side and left-hand side RiemannLiouville fractional integral of order with are defined byrespectively, where is the Gamma function defined as .
It is to be noted that
Reimann integral is reduced as classical integral for .

Definition 8. let /0. A function is said to be -symmetric with respect to if and holds for all .
Following lemma will help us in obtaining our fractional integrals inequalities which can be found in [36].

Lemma 1. Let and is integrable, -symmetric with respect to :(1)If , with .(2)If ,with .

Now, we are ready to develop the Fractional Hermite–Hadamard-type inequalities for interval-valued generalized convex functions.

Theorem 5. Let be generalized convex function and provided is bounded above on and . Then, following fractional integral inequality holds, if and :

Proof. Let be a generalized convex function with and is bounded above by .
Take and .
Since(53) becomesMultiplying both sides of (54) by and then integrating the resulting inequality with respect to over , we obtainBy definition of RiemannLiouville integrable function with , we obtainwhich is the left-hand side of theorem (56).
To prove the right-hand side, we take and :Adding the (57) and (58) and multiplying the resulting inequality with and integrating with respect to over we obtainBy definition of RiemannLiouville integrable function, we getRearranging the above inequality, we get the right-hand side:This completes the proof.

Remark 5. If we put , and , then we will get Hermite–Hadamard-type inequality for fractional function for classical convex function [37].

6. Conclusions

The convex functions and fractional calculus play an important role in applied sciences [38–43]. Here, the new interval-valued generalized convex functions are introduced. By using the notion of interval-valued generalized convex functions and some auxiliary results of interval analysis, some new Hermite–Hadamard- and Fejér-type inequalities are presented. Our results can be considered as generalization of many existing results. Moreover, fractional integral inequality for this generalization is also established.

Data Availability

The data used to support the article are available within the article.

Conflicts of Interest

The authors declare that do not have any conflicts of interest.

Authors’ Contributions

All the authors contributed equally to this paper.

Acknowledgments

This research was supported in part by the Higher Education Commission, Pakistan.

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Copyright © 2020 Kamran et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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