Boundedness of Fractional Integral Operators Containing Mittag-Leffler Function via Exponentially <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 6.59063 8.14084" width="6.59063pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z"/></g></svg>-</span>Convex Functions

Hong, Gang; Farid, G.; Nazeer, Waqas; Akbar, S. B.; Pečarić, J.; Zou, Junzhong; Geng, Shengtao

doi:https://doi.org/10.1155/2020/3584105

Journal of Mathematics

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

Research Article | Open Access

Volume 2020 | Article ID 3584105 | https://doi.org/10.1155/2020/3584105

Boundedness of Fractional Integral Operators Containing Mittag-Leffler Function via Exponentially -Convex Functions

Gang Hong,¹G. Farid,²Waqas Nazeer,³S. B. Akbar,⁴J. Pečarić,⁵Junzhong Zou,⁶and Shengtao Geng⁷

Academic Editor: Sei-Ichiro Ueki

Received29 Jan 2020

Accepted27 Mar 2020

Published07 May 2020

Abstract

The main objective of this paper is to obtain the fractional integral operator inequalities which provide bounds of the sum of these operators at an arbitrary point. These inequalities are derived for -exponentially convex functions. Furthermore, a Hadamard inequality is obtained for fractional integrals by using exponentially symmetric functions. The results of this paper contain several such consequences for known fractional integrals and functions which are convex, exponentially convex, and -convex.

1. Introduction

Convex functions play an important role in many areas of mathematics. They are important especially in the study of optimization problems, theory of inequalities, mathematical analysis, statistical analysis, operation research, and so on. The analytical definition of convex function motivated the authors to define more such functions theoretically; for example, the terms quasi-convex, -convex, -convex, -convex, -convex, and -convex functions [1] are defined by extending or generalizing inequality (1). For this paper, we will use exponentially -convex functions which include exponentially convex, -convex, and convex functions.

Definition 1. A function , where is an interval in , is said to be convex function if the following inequality holds:for all and .

Definition 2 (see [2]). A function , where is an interval in , is said to be exponentially convex function ifholds for all , and . If the inequality in (2) is reversed, then is called exponentially concave.
A generalization of convex function defined on the right half of the real line is the -convex function defined as follows:

Definition 3. (see [3]). Let . A function is said to be -convex function in the second sense ifholds for all and .
It is noted that I-convex function is convex. A further generalization of the above defined functions is given as follows:

Definition 4. (see [4]). Let and be an interval. A function is said to be exponentially -convex in the second sense ifholds for all , and . If the inequality in (4) is reversed, then is called exponentially -concave.
For utilizations of exponentially convex functions, one can see [2, 4–7]. Our aim in this paper is to utilize exponentially -convex functions for establishing bounds of fractional integral operators with kernel Mittag-Leffler function. The Mittag-Leffler function denoted by was introduced by Mittag-Leffler [8] in 1903:where , is the gamma function, and .
The Mittag-Leffler function is a direct generalization of the exponential function to which it reduces for . In the solution of fractional integral and fractional differential equations, it arises naturally. Due to its importance and utilizations, Mittag-Leffler function has been generalized by many authors. By direct involvement in the problems of physics, chemistry, biology, engineering, and other applied sciences, Mittag-Leffler function and its generalizations have successful applications. Recently, in [9], Andrić et al. introduced an extended generalized Mittag-Leffler function which is defined as follows:

Definition 5. Let , , and with , , and . Then, the extended generalized Mittag-Leffler function is defined bywhere is the generalized beta function defined as follows:and is the Pochhammer symbol defined by .
From (4), one can obtain the Mittag-Leffler functions defined in [10–14], see Remark 1.3 of [32].

Lemma 1. (see [9]). If and such that with and , thenFractional integral operators are very useful in mathematical inequalities. The authors have established fractional integral inequalities due to different fractional integral operators, see [15–30]. Many authors have used Mittag-Leffler function to define fractional integral operators. The fractional integral operators corresponding to Mittag-Leffler function (4) are defined as follows:

Definition 6. (see [9]). Let , , and with , , and . Let and . Then, the generalized fractional integral operators containing Mittag-Leffler function are defined byandwhere is the Mittag-Leffler function defined in (6).
From (9) and (10), one can obtain the fractional integral operators defined in [10–14], see Remark 1.4 of [32]. In [20], Farid et al. proved thatandWe will follow the upcoming notations in the main results:The upcoming section contains the main results which address the bounds of fractional integral operators containing Mittag-Leffler functions for exponentially -convex functions. The continuity of such fractional integrals is proved. A Hadamard inequality is established that leads to several Hadamard inequalities for convex, exponentially convex, and -convex functions. Moreover, the results of papers [31, 32] can be obtained in particular.

2. Main Results

Theorem 1. Let be a real-valued function. If is positive and exponentially -convex, then for , the following upper bound for generalized integral operators holds:

Proof. Let . Then, for and , we have the following inequality:As is exponentially -convex, therefore, one can obtainBy multiplying (13) and (14) and then integrating over , we getThe left integral operator follows the upcoming inequality:Now, on the contrary, for and , we have the following inequality:Again from exponentially -convexity of , we haveBy multiplying (20) and (21) and then integrating over , we getThe right integral operator satisfies the following inequality:By adding (19) and (23), required inequality (15) can be obtained.

Corollary 1. If we set in (15), then the following inequality is obtained:

Corollary 2. Along with assumption of Theorem 1, if , then the following inequality is obtained:

Corollary 3. For in (25), we get the following result:

Corollary 4. For in (25), we get the following result for exponentially convex functions:

Theorem 2. With the assumptions of Theorem 1, if , then operators defined in (9) and (10) are bounded and continuous.

Proof. If , then from (19), we havethat is,where . Therefore, is bounded, and also, it is easy to see that it is linear; hence, this is a continuous operator. Also, on the contrary, from (23), one can obtainwhere . Therefore, is bounded and also it is linear, hence continuous.
The next result provides boundedness of the left and the right fractional integrals at an arbitrary point for functions whose derivatives in absolute values are exponentially -convex.

Theorem 3. Let be a real-valued function. If is differentiable and is exponentially -convex, then for , the following fractional integral inequality for generalized integral operators (9) and (10) holds:

Proof. Let and ; by using exponentially -convexity of , we haveFrom (32), one can haveThe product of (16) and (33) gives the following inequality:After integrating the above inequality over , we getThe left-hand side of (35) is calculated as follows:Putting , that is, and also utilizing (8), we haveNow, putting in the second term of the right-hand side of the above equation and then using (9), we getTherefore, (35) takes the following form:Also, from (32), one can haveFollowing the same procedure as we did for (33), one can obtainFrom (39) and (41), we getNow, we let and . Then, by exponentially -convexity of , we haveOn the same lines as we have done for (16), (33), and (40), one can get from (20) and (43) the following inequality:From inequalities (42) and (44), via triangular inequality, (28) can be obtained.
The following results hold as special cases.

Corollary 5. If we put in (28), then the following inequality is obtained:

Definition 7. Let be a function; we say is exponentially symmetric about ifIt is required to give the following lemma which will be helpful to produce Hadamard-type estimations.

Lemma 2. Let be an exponentially -convex function. If is exponentially symmetric, then the following inequality holds:

Proof. For be a closed interval, , and , we haveSince is exponentially -convex, soLet , where . Then, we have , and we getNow, using the fact of exponentially symmetric, we will get (47).

Theorem 4. Let , , be a real-valued function. If is positive, exponentially -convex, and symmetric about , then for , the following fractional integral inequality for generalized integral operators (9) and (10) holds:

Proof. For , we haveAs is exponentially -convex, so for , we haveBy multiplying (52) and (53) and then integrating over , we getfrom which we haveNow, on the contrary, for , we haveBy multiplying (53) and (57) and then integrating over , we getfrom which we haveAdding (56) and (60), we getMultiplying (47) with and integrating over , we getBy using (10) and (14), we getMultiplying (47) with , integrating over , and also using (6) and (10), we getBy adding (63) and (64), we getBy combining (61) and (65), inequality (51) can be obtained.

Corollary 6. If we put in (51), then the following inequality is obtained:

3. Concluding Remarks

We have established the general fractional integral inequalities by using exponentially -convex functions. By selecting particular values at the place of parameters, a variety of results can be obtained. For example, the reader can obtain bounds for fractional integral operators defined by Salim and Faraj in [12] by selecting , bounds for fractional integral operators defined by Rahman et al. in [11] by selecting , bounds for fractional integral operators defined by Shukla and Prajapati in [13] by selecting and , bounds for fractional integral operators defined by Prabhakar in [10] by selecting and , and bounds for Riemann–Liouville fractional integrals by selecting .

Data Availability

All the data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors do not have any conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.

Acknowledgments

This paper was supported by HEC, Pakistan.

References

X. Qiang, G. Farid, J. Pecaric, and S. B. Akbar, “Generalized fractional integral inequalities for exponentially -convex functions,” Journal of Inequalities and Applications, vol. 2020, no. 1, pp. 1–13, 2020.
View at: Publisher Site | Google Scholar
M. U. Awan, M. A. Noor, and K. I. Noor, “Hermite-Hadamard inequalities for exponentially convex functions,” Applied Mathematics & Information Sciences, vol. 12, no. 2, pp. 405–409, 2018.
View at: Publisher Site | Google Scholar
H. Hudzik and L. Maligranda, “Some remarks on -convex functions,” Aequationes Mathematicae, vol. 48, no. 1, pp. 100–111, 1994.
View at: Publisher Site | Google Scholar
N. Mehreen and M. Anwar, “Hermite-Hadamard type inequalities for exponentially -convex functions and exponentially -convex functions in the second sense with applications,” Journal of Inequalities and Applications, vol. 2019, no. 1, p. 92, 2019.
View at: Publisher Site | Google Scholar
G. Alirezaei and R. Mathar, “On exponentially concave functions and their impact in information theory,” in Proceedings of the 2018 Information Theory and Applications Workshop, pp. 11–16, San Diego, CA, USA, February 2018.
View at: Google Scholar
S. Pal and T.-K. L. Wong, “Exponentially concave functions and a new information geometry,” The Annals of Probability, vol. 46, no. 2, pp. 1070–1113, 2018.
View at: Publisher Site | Google Scholar
S. Ullah, G. Farid, K. A. Khan, A. Waheed, and S. Mehmood, “Generalized fractional inequalities for quasi-convex functions,” Advances in Difference Equations, vol. 2019, p. 15, 2019.
View at: Google Scholar
G. M. Mittag-Leffler, “Sur la nouvelle fonction .,” Comptes Rendus Hebdomadaires des Séances de l’ Académie des Sciences Paris, vol. 137, pp. 554–558, 1903.
View at: 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
T. R. Prabhakar, “A singular integral equation with a generalized Mittag-Leffler function in the kernel,” Yokohama Mathematical Journal, vol. 19, pp. 7–15, 1971.
View at: Google Scholar
G. Rahman, D. Baleanu, M. Al Qurashi, S. D. Purohit, S. Mubeen, and M. Arshad, “The extended Mittag-Leffler function via fractional calculus,” The Journal of Nonlinear Sciences and Applications, vol. 10, no. 8, pp. 4244–4253, 2017.
View at: Publisher Site | Google Scholar
T. O. Salim and A. W. Faraj, “A Generalization of Mittag-Leffler function and integral operator associated with integral calculus,” Journal of Fractional Calculus and Applications, vol. 3, no. 5, pp. 1–13, 2012.
View at: Google Scholar
A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler function and its properties,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 797–811, 2007.
View at: Publisher Site | Google Scholar
H. M. Srivastava and Ž. Tomovski, “Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel,” Applied Mathematics and Computation, vol. 211, no. 1, pp. 198–210, 2009.
View at: Publisher Site | Google Scholar
A. O. Akdemir, A. Ekinci, and E. Set, “Conformable fractional integrals and related new integral inequalities,” Journal of Nonlinear and Convex Analysis, vol. 18, no. 4, pp. 661–674, 2017.
View at: Google Scholar
P. Citti, M. G. Cowling, and F. Ricci, “Hardy and uncertainty inequalities on stratified Lie groups,” Advances in Mathematics, vol. 277, pp. 365–387, 2015.
View at: Publisher Site | Google Scholar
M. Dokuyucu, D. Baleanu, and E. Çelik, “Analysis of Keller-Segel model with Atangana-Baleanu fractional derivative,” Filomat, vol. 32, no. 16, pp. 5633–5643, 2018.
View at: Publisher Site | Google Scholar
M. A. Dokuyucu, E. Celik, H. Bulut, and H. M. Baskonus, “Cancer treatment model with the Caputo-Fabrizio fractional derivative,” The European Physical Journal Plus, vol. 133, no. 3, p. 92, 2018.
View at: Publisher Site | Google Scholar
A. Ekinci and M. E. Ozdemir, “Some new integral inequalities via Riemann Liouville integral operators,” Applied and Computational Mathematics, vol. 18, no. 3, pp. 288–295, 2019.
View at: Google Scholar
G. Farid, K. A. Khan, N. Latif, A. U. Rehman, and S. Mehmood, “General fractional integral inequalities for convex and -convex functions via an extended generalized Mittag-Leffler function,” Journal of Inequalities and Applications, vol. 2018, no. 1, p. 243, 2018.
View at: Publisher Site | Google Scholar
M. Kunt and I. Iscan, “Fractional hermite hadamard fejer type inequalities for Ga-convex functions,” Turkish Journal of Inequalities, vol. 2, no. 1, pp. 1–20, 2018.
View at: Google Scholar
A. Kashuri, M. A. Kli, M. Abbas, and H. Budak, “New inequalities for generalized m-convex functions via generalized fractional integral operators and their applications,” International Journal of Nonlinear Analysis and Applications, vol. 10, no. 2, pp. 275–299, 2019.
View at: Google Scholar
A. Kashuri and R. Liko, “Ostrowski type conformable fractional integrals for generalized -preinvex functions,” Turkish Journal of Inequalities, vol. 2, no. 2, pp. 54–70, 2018.
View at: Google Scholar
D. Nie, S. Rashid, A. O. Akdemir, D. Baleanu, and J.-B. Liu, “On some new weighted inequalities for differentiable exponentially convex and exponentially Quasi-convex functions with applications,” Mathematics, vol. 7, no. 8, p. 727, 2019.
View at: Publisher Site | Google Scholar
M. E. Ozdemir, M. Avci-Ardic, and H. Kavurmaci-Onalan, “Hermite Hadamard type inequalities for -convex and -concave functions via fractional integrals,” Turkish Journal of Science, vol. 1, no. 1, pp. 28–40, 2012.
View at: Google Scholar
N. Okur and F. B. Yalcin, “Two dimensional operator harmonically convex functions and related generalized inequalities,” Turkish Journal of Science, vol. 4, no. 1, pp. 30–38, 2019.
View at: Google Scholar
G. Rahman, T. Abdeljawad, F. Jarad, and K. S. Nisar, “Bounds of generalized proportional fractional integrals in general form via convex functions and their applications,” Mathematics, vol. 8, no. 1, p. 113, 2020.
View at: Publisher Site | Google Scholar
S. Rashid, M. A. Noor, and K. I. Noor, “Some new estimates for exponentially -convex functions via extended generalized fractional integral operators,” The Korean Journal of Mathematics, vol. 27, no. 4, pp. 843–860, 2019.
View at: Google Scholar
S. Rashid, F. Safdar, A. O. Akdemir, M. A. Noor, and K. I. Noor, “Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function,” Journal of Inequalities and Applications, vol. 2019, no. 1, p. 299, 2019.
View at: Publisher Site | Google Scholar
H. Yaldiz and A. O. Akdemir, “Katugampola fractional integrals within the class of convex functions,” Turkish Journal of Science, vol. 3, no. 1, pp. 40–50, 2018.
View at: Google Scholar
Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, and S. M. Kang, “Generalized Riemann-Liouville k -Fractional Integrals Associated With Ostrowski Type Inequalities and Error Bounds of Hadamard Inequalities,” IEEE Access, vol. 6, pp. 64946–64953, 2018.
View at: Publisher Site | Google Scholar
S. M. Kang, G. Farid, W. Nazeer, and M. Usman, “Ostrowski Type Fractional Integral Inequalities for Mappings Whose Derivatives are (α, M)-Convex Via Katugampola Fractional Integrals,” Nonlinear Functional Analysis and Applications, vol. 24, pp. 109–126, 2019.
View at: Google Scholar

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Copyright © 2020 Gang Hong 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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