Some New Kinds of Fractional Integral Inequalities via Refined <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M1" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 68.8967 15.2545" width="68.8967pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g><g transform="matrix(.017,0,0,-0.017,15.686,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,22.474,0)"><path id="g113-105" d="M499 88L487 113C459 86 422 65 413 65C403 65 403 74 409 98L461 318C487 426 460 448 431 448C408 448 383 441 352 424C305 398 223 344 157 257H155L243 674C249 702 249 712 240 712C227 712 170 680 87 673L82 648H117C155 648 162 644 150 590L23 -4L30 -12C55 -4 81 3 105 8C113 53 131 150 143 187C199 281 323 387 372 387C390 387 393 369 380 311L328 86C313 19 319 -12 347 -12C379 -12 442 24 499 88Z"/></g><g transform="matrix(.017,0,0,-0.017,35.165,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,49.072,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g><g transform="matrix(.017,0,0,-0.017,62.613,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>-</span>Convex Function

Zahra, Moquddsa; Ashraf, Muhammad; Farid, Ghulam; Nonlaopon, Kamsing

doi:https://doi.org/10.1155/2021/8331092

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest References Copyright Related Articles

Special Issue

Discrete Fractional-Order Systems with Applications in Engineering and Natural Sciences 2021

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 8331092 | https://doi.org/10.1155/2021/8331092

Some New Kinds of Fractional Integral Inequalities via Refined -Convex Function

Moquddsa Zahra,¹Muhammad Ashraf,¹Ghulam Farid,²and Kamsing Nonlaopon³

Academic Editor: Amr Elsonbaty

Received10 Jul 2021

Accepted28 Aug 2021

Published14 Sept 2021

Abstract

In this article, we present new integral inequalities for refined -convex functions using unified integral operators (12) and (13). The established results provide the refinements of several well-known integral and fractional integral inequalities.

1. Introduction

Convex functions are important in diverse fields of mathematics, statistics, engineering, and optimization. Especially in the formation of inequalities, they play a very vital role. In the subject of mathematical analysis, inequalities provide a significant contribution in developing classical concepts and notions. For example, inequalities well known as Cauchy–Schwarz inequality, Chebyshev inequality, Minkowski inequality, Hadamard inequality, and Jensen inequality are utilized frequently in pure and applied mathematics. It is always a challenge to extend, generalize, and refine such inequalities by considering new classes of functions. In this era, researchers are working on classical inequalities concerning fractional integral and derivative operators. It can be observed that the Hadamard inequality is studied more for many kinds of fractional integral and derivative operators than any other classical inequality, see [1–7] for more details.

The aim of this paper is to study the refinements of Hadamard and other integral inequalities recently studied in [8–11]. The consequences of these inequalities also provide refinements of fractional integral inequalities connected with the integral inequalities studied in the recent past.

The article is organized as follows. In Section 2, we suggest some preliminaries. In Section 3, the bounds of unified integral operators are given using refined -convex functions. These are the refinements of bounds already obtained in the literature. In Section 4, some applications of the main results are given in the form of fractional integral inequalities and their refinements.

2. Preliminaries

In this section, we give definitions of different kinds of convex functions and integral operators which will be useful in formulating the results of this paper. Throughout the paper, all the functions are assumed to be real-valued functions until specified.

Definition 1. (see [12]). A function is called convex ifholds for all and .

Definition 2. (see [1]). A function is called -convex if for each , we havewhere and .

Definition 3. (see [13]). A function is called -convex if for each , we havewhere and .

Definition 4. (see [4]). Let is a function with and . A function is said to be -convex, if and for each , we havewhere and .

Definition 5 (see [4]). Let is a function with and . A function is said to be -convex, if and for each , we havewhere and .

Definition 6. (see [14]). Let be a function with and . A function is called refined -convex function, if and for each , we havewhere and .
Inequality (6) gives refinements of several types of convexities when , see [14].
The need for integral operators in the study of fractional derivatives is of immense importance. In the recent era, integral operators are being used extensively for producing new results in the literature. For references, see [2, 4–6]. Next, we give some fundamental integral operators which are used in this paper.

Definition 7. (see [15]). Let and be positive and increasing function having a continuous derivative on . The left and right fractional integrals of with respect to on of order are given bywhere is the gamma function and .

Definition 8. (see [16]). Let and be positive and increasing function having a continuous derivative on . The left and right -fractional integrals of with respect to on of order are given bywhere is the -gamma function and .

Definition 9 (see [17]). Let and , also let with , and , then the generalized fractional integral operators and are defined bywhere is the extended generalized Mittag–Leffler function defined as

Definition 10 (see [16]). Let be real-valued functions defined over with , where is positive and integrable and is differentiable and strictly increasing. Also, let be an increasing function on and , and . Then, for , the left and right integral operators are defined aswhereMittag–Leffler functions give several fractional integrals by assigning particular choice to the parameters involved in it, see Remarks 6 and 7 in [16].

3. Main Results

Throughout the paper, we use the following notation:

Theorem 1. Let be a positive, refined -convex and integrable function defined over . Also, let be an increasing function defined on and be strictly increasing and differentiable function on . Then, for , and , the following result holds:

Proof. For the functions and , the following inequality holds:Using refined -convexity of , one can haveFrom (17) and (18), we have the following integral inequality:Using (12) of Definition 10 on the left side of inequality (19) and making change of the variable by setting on the right-hand side of the above inequality, we obtainThus, we obtainAlso, for and , we can writeandFrom (22) and (23), we have the following integral inequality:Using (13) of Definition 10 on the left-hand side and making change of the variable by setting on the right-hand side of the above inequality, we obtainTherefore,Combining (21) and (26), the required inequality (16) is obtained. Hence, the proof is completed.
Next, we give the refinement of Theorem 1.

Theorem 2. Under the assumptions of Theorem 1, if , then the following result holds:

Proof. From (18) and (23), one can see that, for ,Hence, by following the proof of Theorem 1, one can obtain (27). Hence, the proof is completed.

Corollary 1. Under the assumptions of Theorem 1, (16) gives the following result:

Now, we give the refinement of Theorem 5 in [9] in the following corollary.

Corollary 2. The following inequality for refined -convex function holds:

Proof. Using and in (27), we obtain inequality (30).

Remark 1. (i)For , with , inequality (16) coincides with Theorem 10 in [18](ii)For along with the conditions of (i), inequality (29) coincides with Theorem 6 in [18](iii)For as identity function along with the conditions of (i), inequality (29) coincides with Theorem 5 in [14](iv)For as identity function and along with the conditions of (i), inequality (29) coincides with Theorem 1 in [14](v)For and , inequality (16) coincides with Theorem 4 in [19](vi)For and , inequality (29) coincides with Corollary 1 in [19](vii)For and along with the conditions of (i), inequality (29) coincides with Theorem 3.1 in [20](viii)For along with the conditions of (iv), inequality (29) coincides with Theorem 2 in [14](ix)For and along with the conditions of (iii), inequality (29) coincides with Corollary 8 in [14](x)For and along with the conditions of (iii), inequality (29) coincides with Corollary 14 in [14](xi)For and along with the conditions of (iii), inequality (29) coincides with Corollary 15 in [14](xii)For and along with the conditions of (iii), inequality (29) coincides with Corollary 16 in [14](xiii)For and along with the conditions of (iv), inequality (29) coincides with Corollary 1 in [14](xiv)For and along with the conditions of (iv), inequality (29) coincides with Corollary 2 in [14](xv)For and along with the conditions of (iv), inequality (29) coincides with Corollary 4 in [14](xvi)For and along with the conditions of (iv), inequality (29) coincides with Corollary 5 in [14]By using and making different choices of functions and and the parameters in (16), one can get the refinements of many well-known inequalities for different classes of convex functions which are mentioned in Remark 3 in [9].
Next, we give a lemma which we will use in the proof of upcoming Theorem 3.

Lemma 1. Let be a refined -convex function. If , and , then the following inequality holds:

Proof. Since is refined -convex, then following inequality holds:Using in the above inequality, we obtain (31). This completes the proof.

Remark 2. (i)For and , (31) coincides with Lemma 1 in [19](ii)For , (31) gives refinement of Lemma 1 in [9]

Theorem 3. Under the assumptions of Theorem 1, the following result holds for :

Proof. For the kernel defined in (14) and function , the following inequality holds:Using refined -convexity of , we haveFrom (34) and (35), we have the following integral inequality:Using (13) of Definition 10 on the right-hand side and making change of the variable by setting on the right-hand side of the above inequality, we obtainThe following inequality also holds true for :From (35) and (38), the following integral inequality is obtained:Using (12) of Definition 10 on the left-hand side and making change of the variable on the right-hand side of the above inequality, we obtainNow, using Lemma 1, we can writewhich by using (13) of Definition 10 gives the following integral inequality:Again, using Lemma 1, we can writewhich by using (12) of Definition 10 gives the following fractional integral inequality:Inequality (33) will be obtained by using (37), (40), (42), and (44).
The following theorem is the refinement of Theorem 3.

Theorem 4. Under the assumptions of Theorem 3, if , then the following refinement holds:

Proof. From (35), one can see that, for ,Hence, by following the proof of Theorem 3, one can obtain (45). This completes the proof.

Corollary 3. Under the assumptions of Theorem 3, (33) gives the following result:Now, we give the refinement of Theorem 6 in [9] in the following corollary.

Corollary 4. The following inequality for refined -convex function holds:

Proof. For and in (45), one can obtain (48).

Remark 3. (i)For and , inequality (33) coincides with Theorem 5 in [19](ii)For and , inequality (47) coincides with Corollary 2 in [19]By using and making different choices of functions and and the parameters in (33), one can get the refinements of many well-known inequalities for different classes of convex functions which are mentioned in Remark 5 of [9].

Theorem 5. Let be differentiable functions such that is refined -convex and be strictly increasing over and differentiable over . Also, be an increasing function on and , , and . Then, for , we havewhere

Proof. Using refined -convexity of over impliesAbsolute value property implies the following relation:From (17) and the second inequality of (52), we have the following inequality:which leads to the following fractional integral inequality:Also, inequality (17) and the first inequality of (52) give the following fractional integral inequality:Again, using refined -convexity of over , we can writeandFrom (22) and the second inequality of (57), the following fractional integral inequality is obtained:and (22) and the first inequality of (57) give the following fractional integral inequality:Inequality (49) will be obtained by using (54), (55), (58), and (59). Hence, the proof is completed.
Next, we give refinement of Theorem 5 in the following theorem.

Theorem 6. Under the assumptions of Theorem 5, if , then the following refinement holds:

Proof. From (51), one can see that, for ,Hence, by following the proof of Theorem 5, one can obtain (60). This completes the proof.

Corollary 5. Under the assumptions of Theorem 5, (49) gives the following result:

The following corollary presents the refinement of Theorem 7 in [9].

Corollary 6. The following inequality for refined -convex function holds:

Proof. For and in (60), we obtain (63).

Remark 4. (i)For and , inequality (49) coincides with Theorem 6 in [19](ii)For and , inequality (62) coincides with Corollary 3 in [19]By using and making different choices of functions and and the parameters in (49), one can get the refinements of many well-known inequalities for different classes of convex functions which are mentioned in Remark 6 of [9].

4. Inequalities for Fractional Integral Operators

In this section, we present the bounds of some of the fractional integral operators which will be deduced from the results of Section 3.

Proposition 1. Under the assumptions of Theorem 1, the following result holds:

Proof. For , and in the proof of Theorem 1, we obtain (64).

Proposition 2. Under the assumptions of Theorem 1, the following inequality holds:

Proof. Using as identity function with in the proof of Theorem 1, we obtain the required result.

Corollary 7. For with and , (12) and (13) reduce to the fractional integral operators (8) and (9), which satisfy the following upper bound:

Remark 5. For , (66) gives refinement of Corollary 8 in [9].

Corollary 8. Using and as identity function for along with , (12) and (13) give fractional integral and defined in [15], which satisfy the following upper bound:

Corollary 9. Using and as identity function along with , (12) and (13) reduce to the fractional integral operators and given in [21], which satisfy the following upper bound:

Remark 6. For , (68) gives refinement of Corollary 10 in [9].

Corollary 10. For in Corollary 9, the following upper bound for Riemann–Liouville fractional integral is satisfied:

Remark 7. For , (69) gives refinement of Corollary 11 in [9].
Similar bounds can be obtained for Theorems 3 and 5, which we leave for the reader.

5. Conclusions

This article is about the bounds of unified integral operators via refined -convexity. The obtained results are the refinements of some already published results. Moreover, some deducible fractional integral operators and their related bounds are also given.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

G. A. Anastassiou, “Generalized fractional Hermite-Hadamard inequalities involving -convexity and -convexity,” Series Mathematics and Informatics, vol. 28, no. 2, pp. 107–126, 2013.
View at: Google Scholar
M. Bombardelli and S. Varošanec, “Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1869–1877, 2009.
View at: Publisher Site | Google Scholar
H. Chen and U. N. Katugampola, “Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities for generalized fractional integrals,” Journal of Mathematical Analysis and Applications, vol. 446, no. 2, pp. 1274–1291, 2017.
View at: Publisher Site | Google Scholar
G. Farid, A. U. Rehman, and Q. U. Ain, “-fractional integral inequalities of Hadamard type for -convex functions,” Computational Methods for Differential Equations, vol. 8, no. 1, pp. 119–140, 2020.
View at: Google Scholar
S. Hussain, M. I. Bhatti, and M. Iqbal, “Hadamard-type inequalities for -convex functions,” Jurnal Matematika, vol. 41, pp. 51–60, 2009.
View at: Google Scholar
M. E. Özdemir, M. Avcı, and E. Set, “On some inequalities of Hermite-Hadamard type via -convexity,” Applied Mathematics Letters, vol. 23, pp. 1065–1070, 2010.
View at: Google Scholar
M. E. Özdemir, A. O. Akdemri, and E. Set, “On -convexity and hadamard type inequalities,” Transylvanian Journal of Mathematics and Mechanics, vol. 8, no. 1, pp. 51–58, 2016.
View at: Google Scholar
L. Chen, G. Farid, S. I. But, and S. B. Akbar, “Boundedness of fractional integral operators containing Mittag-Leffler functions,” Turkish Journal of Inequalities, vol. 4, no. 1, pp. 14–24, 2020.
View at: Google Scholar
G. Farid, K. Mahreen, K. Mahreen, and Y.-M. Chu, “Study of inequalities for unified integral operators of generalized convex functions,” Open Journal of Mathematical Sciences, vol. 5, no. 1, pp. 80–93, 2021.
View at: Publisher Site | Google Scholar
G. Farid, W. Nazeer, M. Saleem, S. Mehmood, and S. Kang, “Bounds of Riemann-Liouville fractional integrals in general form via convex functions and their applications,” Mathematics, vol. 6, no. 11, p. 248, 2018.
View at: Publisher Site | Google Scholar
Y. C. Kwun, G. Farid, S. M. Kang, B. K. Bangash, and S. Ullah, “Derivation of bounds of several kinds of operators via (s, m)-convexity,” Advances in Differential Equations, vol. 2020, p. 5, 2020.
View at: Publisher Site | Google Scholar
A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, NY, USA, 1973.
V. G. Mihesan, “A generalization of the convexity, seminar on functional equations, approx. and convex,” 1993.
View at: Google Scholar
G. Farid and M. Zahra, “On fractional integral inequalities for reimann-liouville integrals of refined -convex functions,” 2021.
View at: Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, Netherlands, 2006, North-Holland Mathematics Studies 204.
Y. C. Kwun, G. Farid, S. Ullah, W. Nazeer, K. Mahreen, and S. M. Kang, “Inequalities for a unified integral operator and associated results in fractional calculus,” IEEE Access, vol. 7, pp. 126283–126292, 2019.
View at: Publisher Site | Google Scholar
M. Andrić, G. Farid, and J. Pečarić, “A further extension of Mittag-Leffler function,” Fractional Calculus and Applied Analysis, vol. 21, no. 5, pp. 1377–1395, 2018.
View at: Google Scholar
C. Y. Jung, G. Farid, H. Yasmeen, Y.-P. Lv, and J. Pečarić, “Refinements of some fractional integral inequalities for refined $(\alpha ,h-m)$-convex function,” Advances in Difference Equations, vol. 2021, no. 1, p. 391, 2021.
View at: Publisher Site | Google Scholar
G. Farid and M. Zahra, “Some integral inequalities involving Mittag–Leffler functions for tgs-convex functions,” Computational Methods in Applied Mathematics, vol. 3, no. 5, 2021.
View at: Publisher Site | Google Scholar
M. Tunc, E. Gav, and U. Şanal, “On -convex function and their inequalities,” Series Mathematics and Informatics, vol. 30, no. 5, pp. 679–691, 2015.
View at: Google Scholar
S. Mubeen and G. M. Habibullah, “-fractional integrals and applications,” International Journal of Contemporary Mathematical Sciences, vol. 7, pp. 89–94, 2012.
View at: Google Scholar

Copyright

Copyright © 2021 Moquddsa Zahra 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.

PDF Download Citation

Download other formats

Order printed copies

Views

306

Downloads

420

Citations