Some General Integral Operator Inequalities Associated with <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5268pt" id="M1" height="12.3952pt" version="1.1" viewBox="-0.0657574 -7.8684 10.1561 12.3952" width="10.1561pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-253" d="M559 271C559 395 493 448 433 448C370 448 304 394 275 297C259 245 235 121 219 37C157 54 113 110 113 201C113 310 180 386 276 429L265 457C131 410 23 326 23 186C23 83 86 5 211 -10C180 -159 167 -220 158 -252L165 -261C198 -255 236 -246 251 -218L287 -5C421 9 559 119 559 271ZM484 273C484 144 400 40 293 33L352 331C364 389 382 406 405 406C431 406 484 370 484 273Z"/></g></svg>-</span>Quasiconvex Functions

Kwun, Young Chel; Zahra, Moquddsa; Farid, Ghulam; Agarwal, Praveen; Kang, Shin Min

doi:https://doi.org/10.1155/2021/6696602

Mathematical Problems in Engineering

On this page

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

Special Issue

Various Approaches for Generalized Integral Transforms

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 6696602 | https://doi.org/10.1155/2021/6696602

Some General Integral Operator Inequalities Associated with -Quasiconvex Functions

Young Chel Kwun,¹Moquddsa Zahra,²Ghulam Farid,³Praveen Agarwal,⁴and Shin Min Kang⁵

Academic Editor: HwaJoon Kim

Received21 Nov 2020

Revised15 Dec 2020

Accepted18 Dec 2020

Published08 Jan 2021

Abstract

This paper deals with generalized integral operator inequalities which are established by using -quasiconvex functions. Bounds of an integral operator are established which have connections with different kinds of known fractional integral operators. All the results are deducible for quasiconvex functions. Some fractional integral inequalities are deduced.

1. Introduction and Preliminaries

Convex functions play a vital role in the theory of mathematical analysis. Many generalizations have been given for the convex function, for example, -convex, -convex, -convex, -convex, -convex, -convex, -convex, -convex, and quasiconvex functions (see [1–10]). We will use -quasiconvex functions to study the bounds of unified integral operators, and the established results are directly related with fractional integral operators in particular cases. All the fractional integral operators defined in [11–15] satisfy the results of this paper for -quasiconvex functions, and also the results of [16–19] are reproduced in special cases.

Definition 1. (see [20]). A function is called convex ifholds , where is an interval in .

Definition 2. (see [21]). A function is called -quasiconvex ifholds , where is an interval in and is a bifunction.
For , (2) reduces to quasiconvex function. It is to be noted that every convex function is quasiconvex but converse does not hold.

Example 1. (see [22]). A function defined asis quasiconvex on but not a convex function on the same interval.
The aim of this paper is to establish integral inequalities by using -quasiconvex functions. The results will provide upper bounds of integral operators for -quasiconvex functions, which will behave like compact formulas that unify bounds of various kinds of operators already defined in literature. Next, we give some generalized fractional integral operators connected with the findings of this work.

Definition 3. (see [14]). Let and be positive and increasing function on , and also, let have continuous derivative on . The left and right fractional integrals of with respect to on of order , where , are given as follows:where

Definition 4. (see [23]). Let and be positive and increasing function on , and also let have continuous derivative on . The left and right -fractional integrals of with respect to on of order , where , are given as follows:where

Definition 5. (see [24]). Let and ; also, let , , with , , and , then the generalized fractional integral operators and are defined aswhere is given byIn [11], Farid defined a unified integral operator and proved the boundedness, linearity, and continuity of these integrals. It is given in the following definition.

Definition 6. (see [11]). Let where be the functions such that f is positive and integrable over and is differentiable and strictly increasing. Also, let be an increasing function on and , , 0 and . Then, for , the left and right integral operators are defined aswhereBy making particular choices for and parameters involved in (9), several fractional integrals can be obtained (see [19], Remarks 6 and 7). In [19], Zhao. et al. proved the bounds of unified integral operators for quasiconvex functions stated in Theorems 1 to 4.

Theorem 1. Consider be a positive quasiconvex function and be differentiable and strictly increasing function. Also, be an increasing function on and , 0, , and . Then, for , we have

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

Theorem 3. Along with the assumptions of Theorem 1, if , then the following result holds:

Theorem 4. Consider be two differentiable functions such that is quasiconvex and be strictly increasing for . Also, be an increasing function on and , 0 and and . Then, for , we havewhereAll of the above results are direct consequences of the results of this paper. Also, some of the results in papers [16–18] are very special cases. In the next section, -quasiconvexity has been used frequently to obtain the upper bounds and the Hadamard inequality, which gives upper as well as lower bounds of unified integral operators. Also defining convolution of two functions, some bounds have been obtained for -quasiconvexity of of differential function . In Section 3, some applications of the main results are given. In the whole paper, we use the notation

2. Main Results

Theorem 5. Consider is -quasiconvex and positive and differentiable and strictly increasing functions. If is increasing function on and , , and , then for , the following inequality holds:

Proof. For the kernel defined in (12) and the function , we can write the following inequality:By using -quasiconvexity of on , one can getThe following integral inequality is constituted from (21) and (22):Using (10) in the left and integrating on the right side of inequality (23), we obtain the following upper bound of the left integral operator:Now, following the similar technique for and , we can writeUsing -quasiconvexity for and , we obtainThe following integral inequality is constituted from (25) and (26):Using (11) in the left and integrating on the right side of the above inequality, we obtain the following upper bound of the right integral operator:By summing (24) and (28), the inequality (20) can be obtained.

Corollary 1. Using in (20), we get the following result:

Remark 1. (i)For in (20), we obtain inequality (13) of Theorem 1.(ii)For , for the left-hand integral and for the right-hand integral in (20) with , we obtain Theorem 2.1 in [17].(iii)For in the resulting inequality of (ii), we obtain Corollary 2.2 in [17].(iv)For , for the left-hand integral and for the right-hand integral in (20) with , we obtain Corollary 2.3 in [17].(v)Under the same assumptions as in (ii) along with as identity function, the result (20) reduces to Corollary 2.4 in [17].(vi)Under the same assumptions as in (iv) along with as identity function, the result (20) reduces to Corollary 2.5 in [17].(vii)Under the same assumptions as in (ii), if is increasing on , the result (20) reduces to Corollary 2.6 in [17].(viii)Under the same assumptions as in (ii), if is decreasing on , the result (20) reduces to Corollary 2.7 in [17].(iix)For in the resulting inequality of (viii), we obtain Corollary 2.2 in [18].

Theorem 6. The following result holds under the suppositions of Theorem 5:

Proof. Using in (24) and in (28) and then adding the obtained inequalities, we get (30).

Corollary 2. Using in (30), we get the following result:

Remark 2. (i)For in (30), we obtain inequality (14) of Theorem 2.(ii)For , for the left-hand integral and for the right-hand integral in (30) with , we obtain Theorem 3.1 in [17].(iii)For , for the left-hand integral and for the right-hand integral in (31) with , we obtain Corollary 3.2 in [17].(iv)For , replacing with , for the left-hand integral, for the right-hand integral, and as identity function in (30), we obtain Theorem 2.1 in [18].(v)Under the same assumptions as in (iv) along with , the result (30) reduces to Theorem 3.3 in [16].Before proceeding to the next result, we will prove the following lemma. This lemma is necessary to prove the upcoming result.

Lemma 1. Let be -quasiconvex function. If , then the following inequality holds:

Proof. Using -quasiconvexity of the function , the upcoming inequality holds:Using in above inequality, we get the required inequality.

Remark 3. Using , (32) coincides with Lemma 1 in [19].

Theorem 7. Along with the assumptions of Theorem 5, if and , then the following results hold:provided or .

Proof. From the kernel defined in (12) and the function , we can writeUsing -quasiconvexity of on , we haveThe following inequality is constituted from (36) and (37):Using (11) in the left and integrating on the right side of the above inequality, we obtain the following upper bound of the right integral operator:Also,From (37) and (40), we getUsing (10) in the left and integrating on the right side of the above inequality, we obtain the following upper bound of the left integral operator:Now using (32) of Lemma 1, we can have

Case 1. If , then by using (11), in (43), and integrating over , we getIn this case, we also have

Case 2. If , then by using (11) in (43), we getIn this case, we also haveThe inequality (34) will be obtained by summing (39) with (42) and (44) with (45) and then combining the resulting inequalities. The inequality (35) will be obtained by summing (39) with (42) and (46) with (47) and then combining the resulting inequalities.

Corollary 3. For in (34) and (35), we get the following results:

Remark 4. (i)For in (34), we get inequality (15) of Theorem 3.(ii)For , for the left-hand integral and for the right-hand integral in (34) with , we obtain Theorem 2.16 in [17].(iii)For , for the left-hand integral and for the right-hand integral in (48) with , we obtain Corollary 2.17 in [17].(iv)For , for the left-hand integral and for the right-hand integral in (34) with , we obtain Corollary 2.18 in [17].(v)Under the same assumptions as in (ii) along with as identity function, the result (34) reduces to Corollary 2.19 in [17].(vi)Under the same assumptions as in (iv) along with as identity function, the result (34) reduces to Corollary 2.20 in [17].(vii)Under the same assumptions as in (ii), if is increasing on , the result (34) reduces to Corollary 2.21 in [17].(viii)Under the same assumptions as in (ii), if is decreasing on , the result (34) reduces to Corollary 2.22 in [17].

Theorem 8. Consider are two differentiable functions such that is -quasiconvex and is strictly increasing for . If is increasing function on and , , and , then for , the following inequality holds:where and are defined in (17) and (18).

Proof. The -quasiconvexity of implies the following inequality:which is equivalent toFirst considerThe following inequality is constituted from (21) and (53):from which we getNow, we considerUsing (21) and (56), we getNow, again using -quasiconvexity of , we haveSimilarly using (25) and (58), one can obtainThe inequality (50) will be obtained by summing (55), (57), (59), and (60).

Corollary 4. For in (50), we get the following result:

Remark 5. (i)For in (50), we obtain inequality (16) of Theorem 4.(ii)For , for the left-hand integral and for right-hand integral in (50) with , we obtain Theorem 2.8 in [17].(iii)For , for the left-hand integral and for the right-hand integral in (61) with , we obtain Corollary 2.9 in [17].(iv)For , for the left-hand integral and for the right-hand integral in (50) with , we obtain Corollary 2.10 in [17].(v)Under the same assumptions as in (ii) along with as identity function, the result (50) reduces to Corollary 2.11 in [17].(vi)Under the same assumptions as in (iv) along with as identity function, the result (50) reduces to Corollary 2.12 in [17].(vii)Under the same assumptions as in (ii), if is increasing on , the result (50) reduces to Corollary 2.13 in [17].(viii)Under the same assumptions as in (ii), if is decreasing on , the result (50) reduces to Corollary 2.14 in [17].(ix)Under the same assumptions as in (ii), if in addition we put and in the left- and right-hand integrals, respectively, we obtain Theorem 3.2 in [17].(x)For in the resulting inequality of (ix), we obtain Corollary 3.5 in [17].(xi)For in the resulting inequality of (x), we obtain Corollary 3.6 in [17].

3. Applications

In this section, we present some results by applying theorems of previous section.

Proposition 1. The following result holds under the suppositions of Theorem 5:

Proof. For and , with is increasing for in the proof of Theorem 5 we get (62).

Proposition 2. The following result holds under the suppositions of Theorem 5:

Proof. Using as identity function, , and in the proof of Theorem 5, we get inequality (63).

Corollary 5. For in Theorem 5, the following bound for is satisfied:

Corollary 6. Using in (63), fractional integrals and defined in [14] are obtained which satisfy the following bound:

Corollary 7. Using in (63), fractional integral operators and given in [26] are obtained which satisfy the following bound:

Corollary 8. Using , and , in (10) and (11), respectively, with , then fractional integral operators and given in [27] are obtained which satisfy the following bound:

Corollary 9. Using , and , in (10) and (11), respectively, with , then fractional integral operators and are obtained which satisfy the following bound:

Corollary 10. Using and in (10) and (11), respectively, with , then fractional integral operators and given in [28] are obtained which satisfy the following bound:

Corollary 11. Using , , in (10) and (11), respectively, with , fractional integral operators and are obtained given in [13] which satisfy the following bound:

Corollary 12. Using , in (10), and in (11), where with , then following fractional integral operators are obtained given in [12]:Furthermore, the following bound is also satisfied:

Corollary 13. For and in (10) and in (11), where with , then the following fractional integral operators are obtained given in [29]:Furthermore, the following bound is also satisfied:Similar bounds can be obtained for Theorems 7 and 8 which we leave for the reader.

4. Concluding Remarks

A notion namely -quasiconvexity is studied under an integral operator that associates with different kinds of operators independently defined by various authors during the last two decades. The consequences of the results are compiled in the form of corollaries and remarks. Although some of the particular cases are analyzed in Section 3 by applying Theorem 5, the reader can further compute more results as desired by applying other theorems.

Data Availability

There are no additional data required for the finding of results of this paper.

Disclosure

There is no funding available for the publication of this paper.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

All authors have equal contribution in this article.

Acknowledgments

This work was supported by the Dong-A University Research Fund.

References

G. Adilov and I. Yesilce, “On generalization of the concept of convexity,” Hacettepe Journal of Mathematics and Statistics, vol. 41, no. 5, pp. 723–730, 2012.
View at: Google Scholar
G. Adilov and I. Yesilce, “B⁻¹-Convex sets and B⁻¹-Measurable maps,” Numerical Functional Analysis and Optimization, vol. 33, no. 2, pp. 131–141, 2012.
View at: Publisher Site | Google Scholar
G. A. Anastassiou, “Generalized fractional Hermite-Hadamard inequalities involving -convexity and (s,m)-convexity,” Series: Mathematics and Informatics, vol. 28, no. 2, pp. 107–126, 2013.
View at: Google Scholar
W. Briec and C. Horvath, “B-Convexity,” Optimization, vol. 53, no. 2, pp. 103–127, 2004.
View at: Publisher Site | Google Scholar
M. Bombardelli and S. Varošanec, “Properties of -convex functions related to the Hermite-Hadamard-Fejér inequalities,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1869–1877, 2009.
View at: Google Scholar
H. Hudzik and L. Maligranda, “Some remarks on s-convex functions,” Aequationes Mathematicae, vol. 48, no. 1, pp. 100–111, 1994.
View at: Publisher Site | Google Scholar
Y. C. Kwun, M. Zahra, G. Farid, S. Zainab, and S. M. Kang, “On a unified integral operator for -convex functions,” Advances in Difference Equations, vol. 297, 2020.
View at: Google Scholar
V. G. Mihesan, “A generalization of convexity,” in Proccedings of the Seminar on Functional Equations, Approx. And Convex, Cluj-Napoca, Romania, 1993.
View at: Google Scholar
M. E. Özdemir, A. O. Akdemri, and E. Set, “On (h,m)-convexity and Hadamard type inequalities,” Journal of Mathematics and Mechanics, vol. 8, no. 1, pp. 51–58, 2016.
View at: Google Scholar
S.-M. Yuan and Z.-M. Liu, “Some properties of α-convex and α-quasiconvex functions with respect to n-symmetric points,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1142–1150, 2007.
View at: Publisher Site | Google Scholar
G. Farid, “Existence of an integral operator and its consequences in fractional and conformable integrals,” Open Journal of Mathematical Sciences, vol. 3, no. 1, pp. 210–216, 2019.
View at: Publisher Site | Google Scholar
F. Jarad, E. Ugurlu, T. Abdeljawad, and D. Baleanu, “On a new class of fractional operators,” Advances in Difference Equations, vol. 2017, p. 247, 2017.
View at: Google Scholar
T. U. Khan and M. A. Khan, “Generalized conformable fractional operators,” Journal of Computational and Applied Mathematics, vol. 346, pp. 378–389, 2019.
View at: Publisher Site | Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory And Applications Of Fractional Differential Equations, vol. 204, Elsevier, New York, NY, USA, 2006.
G. Rahman, D. Baleanu, M. A. Qurashi, S. D. Purohit, S. Mubeen, and M. Arshad, “The extended Mittag-Leffeler function via fractional calculus,” Journal of Nonlinear Science and Applications, vol. 10, pp. 4244–4253, 2013.
View at: Google Scholar
S. S. Dragomir and C. E. M. Pearce, “Quasi-convex functions and Hadamard's inequality,” Bulletin of the Australian Mathematical Society, vol. 57, no. 3, pp. 377–385, 1998.
View at: Publisher Site | Google Scholar
G. Farid, C. Y. Jung, S. Ullah, W. Nazeer, M. Waseem, and S. M. Kang, “Some generalized -fractional integral inequalities for quasi-convex functions,” Journal of Computational Analysis and Applications, vol. 29, no. 3, pp. 454–467, 2021.
View at: Google Scholar
S. Ullah, G. Farid, K. A. Khan, A. Waheed, and S. Mehmood, “Generalized fractional inequalities for quasi-convex functions,” Advance Difference Equations, vol. 15, 2019.
View at: Google Scholar
D. Zhao, G. Farid, M. Zeb, S. Ahmad, and K. Mahreen, “On boundedness of unified integral operators for quasiconvex functions,” Advance Difference Equations, vol. 38, 2020.
View at: Google Scholar
A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, NY, USA, 1973.
M. E. Gordji, M. R. Delavar, and M. D. L. Sen, “On φ-convex functions,” Journal of Mathematical Inequalities, vol. 10, no. 1, pp. 173–183, 2016.
View at: Publisher Site | Google Scholar
D. A. Ion, “Some estimates on the Hermite–Hadamard inequality through quasi-convex functions,” Analele Universitatii din Craiova. Seria Matematica, vol. 34, pp. 82–87, 2007.
View at: Google Scholar
Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, and S. M. Kang, “Generalized Riemann-Liouville -fractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities,” IEEE Access, vol. 6, 2018.
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
Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, K. Mahreen, and S. M. Kang, “Inequalities for a unified integral operator and associated results in fractional integrals,” IEEE Access, vol. 7, 2019.
View at: Publisher Site | Google Scholar
S. Mubeen and G. M. Habibullah, “κ-fractional integral and application,” International Journal of Contemporary Mathematical Sciences, vol. 7, pp. 89–94, 2012.
View at: 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
M. Z. Sarikaya, Z. Dahmani, M. E. Kiris, and F. Ahmad, “(k, s)-Riemann-Liouville fractional integral and applications,” Hacettepe Journal of Mathematics and Statistics, vol. 1, no. 45, pp. 77–89, 2016.
View at: Publisher Site | Google Scholar
S. Habib, S. Mubeen, and M. N. Naeem, “Chebyshev type integral inequalities for generalized -fractional conformable integrals,” Journal of Inequalities and Special Functions, vol. 9, no. 4, pp. 53–65, 2018.
View at: Google Scholar

Copyright

Copyright © 2021 Young Chel Kwun 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

270

Downloads

596

Citations