Some New Refinements of Hermite–Hadamard-Type Inequalities Involving <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 17.4157 13.4804" width="17.4157pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-245" d="M642 419L635 448C586 435 552 411 532 392C518 379 509 351 496 304C484 260 462 196 445 150C424 93 388 40 314 30L413 508L406 510L344 491L257 36C205 46 175 84 175 169C175 195 180 241 185 277C189 304 190 331 190 358C190 413 175 448 141 448C111 448 69 429 23 373L30 347C51 365 68 375 81 375C93 375 108 368 108 327C108 307 104 268 101 239C98 211 95 180 95 155C95 33 159 -8 248 -14L213 -186C206 -220 221 -254 230 -261L261 -230L306 -11C339 -4 366 6 389 20C431 46 473 87 503 155C533 224 557 286 571 325C586 367 606 393 642 419Z"/></g><g transform="matrix(.012,0,0,-0.012,10.45,4.134)"><path id="g50-108" d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 405 485 418Z"/></g></svg>-</span>Riemann–Liouville Fractional Integrals and Applications

Awan, Muhammad Uzair; Talib, Sadia; Chu, Yu-Ming; Noor, Muhammad Aslam; Noor, Khalida Inayat

doi:https://doi.org/10.1155/2020/3051920

Mathematical Problems in Engineering

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Abstract Introduction and Preliminaries Applications Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Recent Trends in Special Functions and Analysis of Differential Equations

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Volume 2020 | Article ID 3051920 | https://doi.org/10.1155/2020/3051920

Some New Refinements of Hermite–Hadamard-Type Inequalities Involving -Riemann–Liouville Fractional Integrals and Applications

Muhammad Uzair Awan,¹Sadia Talib,¹Yu-Ming Chu,²Muhammad Aslam Noor,³and Khalida Inayat Noor³

Guest Editor: Praveen Agarwal

Received18 Feb 2020

Accepted31 Mar 2020

Published25 Apr 2020

Abstract

The main objective of this article is to establish some new fractional refinements of Hermite–Hadamard-type inequalities essentially using new -Riemann–Liouville fractional integrals, where . Using this new fractional integral, we also derive two new fractional integral identities. Applications of the obtained results are also discussed.

1. Introduction and Preliminaries

Let be a convex function; then,

The above inequality is known as Hermite–Hadamard’s inequality [1–5]. This inequality provides us a necessary and sufficient condition for a function to be convex. It can be considered as one of the most extensively studied results pertaining to convexity. Since the appearance of this result in the literature, it gained popularity, and many new generalizations for this classical result have been obtained. This can be attributed to its applications in various other fields such as in numerical analysis and in mathematical statistics. For more details on generalizations of convexity, Hermite–Hadamard-like inequalities, and its applications, see [6–14].

Fractional calculus is a calculus in which we study about the integrals and derivatives of any arbitrary real or complex order. The history of fractional calculus is not very much old, but in the short span of time, it experienced a rapid development. Recently, the generalizations [15–25], extensions [26–32], and applications [33–46] for fractional calculus have been made by many researchers. The Riemann–Liouville fractional integrals are defined as follows.

Definition 1. (see [47]). Let . Then, Riemann–Liouville integrals and of order with are defined bywhereis the well-known gamma function.
Sarikaya et al. [10] elegantly utilized this concept in establishing fractional analogue of Hermite–Hadamard’s inequality. This idea motivated other researchers, and consequently, many new generalizations of Hermite–Hadamard’s inequality have been obtained using the concept of Riemann–Liouville fractional integrals.
Sarikaya and Karaca [12] introduced -analogue of Riemann–Liouville fractional integrals and discussed some of its basic properties. They defined this concept in the following way: to be more precise, let be piecewise continuous on and integrable on any finite subinterval of . Then, for , we consider -Riemann–Liouville fractional integral of of order asIf , then -Riemann–Liouville fractional integrals reduce to classical the Riemann–Liouville fractional integral. It is worth to mention here that the concept of the -Riemann–Liouville fractional integral is a significant generalization of Riemann–Liouville fractional integrals; as for , the properties of -Riemann–Liouville fractional integrals are quite different from the classical Riemann–Liouville fractional integrals.
Another important generalization of Riemann–Liouville fractional integrals is -Riemann–Liouville fractional integrals.

Definition 2. (see [6]). Let be a finite interval of the real line and . Also, let be an increasing and positive monotone function on , having a continuous derivative on . Then, the left- and right-sided -Riemann–Liouville fractional integrals of a function with respect to another function on are defined asrespectively; is the gamma function.
For some recent research works, see [48].
Recently, Liu et al. [14] obtained some interesting results pertaining to Hermite–Hadamard’s inequality involving -Riemann–Liouville fractional integrals. Motivated by the research work of Liu et al. [14], we obtain some new refinements of fractional Hermite–Hadamard’s inequality essentially using -Riemann–Liouville fractional integrals. We also discuss applications of the obtained results to means. We show that our results represent significant generalization of some previous results.

2. Hermite–Hadamard’s Inequality

In this section, we derive a new refinement of Hermite–Hadamard’s inequality via the -Riemann–Liouville fractional integral.

Definition 3. Let , be a finite interval of the real line , and . Also, let be an increasing and positive monotone function on , having a continuous derivative on . Then, the left- and right-sided -Riemann–Liouville fractional integrals of a function with respect to another function on are defined asrespectively;is the -analogue of gamma function.
The -analogues of beta function and incomplete beta function are, respectively, defined asWe now derive the main result of this section.

Theorem 1. Let and be a positive function and . Also, suppose that is a convex function on , is an increasing and positive monotone function on , having a continuous derivative on , and . Then, for , the following -fractional integral inequalities hold:

Proof. Using the convexity of , we haveMultiplying both sides by and then integrating with respect to on , we haveNow, making the substitution , we haveAlso, using the convexity property of , we haveMultiplying both sides by and then integrating it with respect to on , we obtainThis impliesThe proof is completed.

3. Some More Fractional Inequalities of Hermite–Hadamard Type

We now derive two new fractional integral identities involving -Riemann–Liouville fractional integrals. These results will serve as auxiliary results for obtaining our next results.

Lemma 1. Let and be a differentiable mapping on . Also, suppose that , is an increasing and positive monotone function on , having a continuous derivative on , and . Then, for , the following identity holds:

Proof. Consider and .
Now,Similarly,It follows that

Example 1. Let . Then, all the assumptions in Lemma 1 are satisfied. Observe that .This impliesAlso,

Example 2. Let . Then, all the assumptions in Lemma 1 are satisfied. Observe that .This impliesAlso,

Lemma 2. Let and be a differentiable mapping on . Also, suppose that , is an increasing and positive monotone function on , having a continuous derivative on , and . Then, for , the following identity holds:where

Proof. SupposeSumming , and , we get the required result.

Example 3. Let . Then, all the assumptions in Lemma 2 are satisfied. Note that .This impliesAlso,where is defined in Lemma 2.This implies

Example 4. Let . Then, all the assumptions in Lemma 2 are satisfied. Note that .This impliesAlso,where is defined in Lemma 2.This impliesBefore proceeding to next results, let us recall the definition of -convex function of Breckner type.

Definition 4. (see [49]). A function is said to be -convex function of Breckner type if

Theorem 2. Let and be a differentiable mapping on . Also, suppose that is Breckner type of -convex on , is an increasing and positive monotone function on , having a continuous derivative on , and . Then, for , the following inequality holds:whererespectively.

Proof. Using Lemma 1 and the fact that is Breckner type of -convex function, we havewhereThis completes the proof.

Theorem 3. Let be a differentiable function on with . Also, suppose that is Breckner type of -convex function. If is an increasing and positive monotone function on , having a continuous derivative on and , then for , the following inequality holds:where and are given by (44) and (45), respectively.

Proof. Using Lemma 2, the property of modulus, and the given hypothesis of the theorem, we haveUsing substitution and the fact that is Breckner type of -convex function, we havewhere and are given by (44) and (45), respectively. AndThis completes the proof.

4. Applications

In this section, we discuss some applications of Theorem 2 to means by considering a particular example of -convexity. First of all, we recall some previously known concepts related to means [50].

For arbitrary real numbers , , we define the following:(1)Arithmetic mean:(2)Logarithmic mean:(3)Generalized log-mean:

We now give the main results of this section.

Proposition 1. Let with ; then,whererespectively.

Proof. Applying Theorem 2 for , , and , we obtain the required result.

Proposition 2. Let with ; then,where and are given by (57) and (58), respectively.

Proof. Applying Theorem 3 for , , and , we obtain the required result.

5. Conclusion

In this article, we obtain some new fractional estimates of Hermite–Hadamard’s inequality essentially using a new -analogue of -fractional integrals. We derive two new fractional integral identities in the setting of -fractional calculus. In order to check the validity of these identities, we discuss some particular examples. In the final section, we have discussed applications of Theorems 2 and 3 to means.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant nos. 61673169, 11701176, 11626101, and 11601485).

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Copyright © 2020 Muhammad Uzair Awan 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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