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Guo, Shuya; Chu, Yu-Ming; Farid, Ghulam; Mehmood, Sajid; Nazeer, Waqas

doi:https://doi.org/10.1155/2020/2410385

Journal of Function Spaces

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

Research Article | Open Access

Volume 2020 | Article ID 2410385 | https://doi.org/10.1155/2020/2410385

Fractional Hadamard and Fejér-Hadamard Inequalities Associated with Exponentially -Convex Functions

Shuya Guo,¹Yu-Ming Chu,^2,3Ghulam Farid,⁴Sajid Mehmood,⁵and Waqas Nazeer⁶

Academic Editor: Hugo Leiva

Received10 Apr 2020

Accepted11 Jul 2020

Published07 Aug 2020

Abstract

The aim of this paper is to present the fractional Hadamard and Fejér-Hadamard inequalities for exponentially -convex functions. To establish these inequalities, we will utilize generalized fractional integral operators containing the Mittag-Leffler function in their kernels via a monotone function. The presented results in particular contain a number of fractional Hadamard and Fejér-Hadamard inequalities for -convex, -convex, -convex, exponentially convex, exponentially -convex, and convex functions.

1. Introduction and Preliminaries

Integral operators play an important role in the subject of mathematical analysis. Fractional integral operators have been proven very useful in almost all fields of science and engineering. By using fractional integral operators, a lot of well-known inequalities have been studied, and in the consequent, they are generalized and extended in the subject of fractional calculus. Fractional integral inequalities provide support in the formation of modeling of physical phenomenons. They also provide their role in the uniqueness of solutions of fractional boundary value problems. A large number of fractional integral inequalities exist in literature due to fractional integral operators (see, [1–5]). The Riemann-Liouville fractional integral operators are the first formulation of fractional integral operators of nonintegral order.

Definition 1 [6]. Let . The Riemann-Liouville fractional integral operators and of order are defined by where .

Next, we give the definition of generalized fractional integral operators containing the Mittag-Leffler function in their kernels as follows.

Definition 2 [7]. Let , , with , , and . Let and . Then, the generalized fractional integral operators and are defined by where is the generalized Mittfag-Leffler function defined as follows:

Lemma 3 [7]. If , , with , and , then

Recently in [8], Farid defined the following unified integral operators.

Definition 4. Let , be the functions such that be a positive and integrable and be a differentiable and strictly increasing. Also, let be an increasing function on and , , and . Then, for , the integral operators and are defined by

If we take in (7) and (9), then we get the following generalized fractional integral operators containing the Mittag-Leffler function.

Definition 5. Let , be the functions such that be a positive and integrable and be a differentiable and strictly increasing. Also, let , , and . Then, for , the integral operators and are defined by

Remark 6. The operators (9) and (10) are the generalization of the following fractional integral operators: (i)By taking and , the Riemann-Liouville fractional integral operators (1) and (3) can be achieved(ii)By taking , the fractional integral operators (5) and (6) can be achieved(iii)By taking and , the fractional integral operators defined by Salim-Faraj in [9] can be achieved(iv)By taking and , the fractional integral operators defined by Rahman et al. in [10] can be achieved(v)By taking , , and , the fractional integral operators defined by Srivastava-Tomovski in [11] can be achieved(vi)By taking , , and , the fractional integral operators defined by Prabhakar in [12] can be achieved

From generalized fractional integral operator (9), we have

Hence,

Similarly, from the generalized fractional integral operator (10), we have

In this paper, we will use the following notations frequently:

Convex functions are also very important in the field of mathematical analysis (see, [13–15]).

A real-valued function is said to be convex on ( is an interval in ), if holds, for all , and . The function is said to be concave if the reverse of inequality (16) holds.

Convex functions are further extended and generalized in various ways by using different techniques. For example in [16], Qiang et al. introduced exponentially -convex function which is a generalization of -convex, -convex, -convex, exponentially convex, exponentially -convex, and convex functions.

Recently, is as follows.

Definition 7. Let and be an interval. A function is said to be exponentially -convex on , if holds, for all , , and .

Remark 8. (i)If we take in (17), then exponentially -convex function defined by Mehreen and Anwar in [17] can be achieved(ii)If we take in (17), then exponentially convex function defined by Awan et al. in [18] can be achieved(iii)If we take in (17), then -convex function defined by Efthekhari in [19] can be achieved(iv)If we take and in (17), then -convex function defined by Hudzik and Maligranda in [20] can be achieved(v)If we take and in (17), then -convex function defined by Toader in [21] can be achieved(vi)If we take and in (17), then convex function (16) can be achieved

A convex function is also equally defined by the well-known Hadamard inequality stated as follows: where is a convex function.

In [22], Fejér gave a generalization of the Hadamard inequality stated as follows: where is a convex function and is positive, integrable, and symmetric to .

The inequality (19) is well known as the Fejér-Hadamard inequality. The Hadamard and the Fejér-Hadamard inequalities have been analyzed by many authors and produced frequently their generalizations, refinements, and extensions (see, [23–37]).

The main purpose of this article is to establish the Hadamard and the Fejér-Hadamard inequalities for exponentially -convex functions by utilizing the generalized fractional integral operators (9) and (10) containing the Mittag-Leffler function. These inequalities lead to produce results for several kinds of convexities and fractional integral operators composed in Remark 8 and Remark 6, respectively. In Section 2, we prove the Hadamard inequalities for generalized fractional integral operators (9) and (10) via exponentially -convex functions. In Section 3, we prove the Fejér-Hadamard inequalities for these generalized fractional integral operators via exponentially -convex functions. Also, it is shown that these inequalities are generalizations of the Hadamard and the Fejér-Hadamard inequalities proved in [27–30, 32, 33, 35–37]. In the whole paper, we will consider real parameters of the Mittag-Leffler function.

2. Hadamard Inequalities

Here, we will give two versions of the generalized fractional Hadamard inequality.

Theorem 9. Let , be the real-valued functions. If be a integrable and exponentially -convex and be a differentiable and strictly increasing. Then, for integral operators (9) and (10), the following inequalities hold: where for , for and .

Proof. Since is exponentially -convex function on , for , we have the following inequalities:

Multiplying both sides of (21) with and integrating over , we have

Putting and in (23), we get

By using (9), (10), and (14), the first inequality of (20) is obtained.

Now, multiplying both sides of (22) with and integrating over , we have

First, we calculate the following integrals of (25): as for . Hence,

By using (9), we have

Now, putting , , and using (27), (28) in (25), the second inequality of (20) is obtained.

Special cases of the above generalized fractional Hadamard inequality are highlighted in the following remark.

Remark 10. (i)If we take in (20), then Hadamard inequality for the exponentially -convex function of second kind is obtained(ii)If we take in (20), then Hadamard inequality for the exponentially convex function is obtained(iii)If we take , , and in (20), then [35] (Theorem 2.1) is obtained(iv)If we take , , and in (20), then [35] (Theorem 3.1) is obtained(v)If we take , , , and in (20), then [27] (Theorem 2.1) is obtained(vi)If we take , , , and in (20), then [28] (Theorem 3) is obtained(vii)If we take , , , and in (20), then [36] (Theorem 2) is obtained(viii)If we take , , , and in (20), then [30] (Theorem 2.1) is obtained(ix)If we take , , , , and in (20), then classical Hadamard inequality is obtained

In the following, we give another version of the Hadamard inequality for generalized fractional integral operators.

Theorem 11. Let , be the real-valued functions. If be a integrable and -convex and be a differentiable and strictly increasing. Then, for integral operators (9) and (10), the following inequalities hold: where for , for and .

Proof. Since is exponentially -convex function on , for , we have the following inequalities:

Multiplying both sides of (30) with and integrating over , we have

Putting and in (32), we get

By using (9), (10), and (14), the first inequality of (29) is obtained.

Now, multiplying both sides of (31) with and integrating over , we have

Putting and in (34), then by using (9), (10), and (27), the second inequality of (29) is obtained.

The following remark establishes the connection with already published results.

Remark 12. (i)If we take in (29), then Hadamard inequality for the exponentially -convex function of second kind is obtained(ii)If we take in (29), then Hadamard inequality for the exponentially convex function is obtained(iii)If we take , , and in (29), then [32] (Theorem 2.11) is obtained(iv)If we take , , , and in (29), then [29] (Theorem 3.10) is obtained(v)If we take , , , and in (29), then [37] (Theorem 4) is obtained(vi)If we take , , , , and in (29), then classical Hadamard inequality is obtained

3. Fejér-Hadamard Inequalities

Here, we give two versions of the Fejér-Hadamard inequality.

Theorem 13. Let , be the real-valued functions. If be a integrable, exponentially -convex and and be a differentiable and strictly increasing. Also, let be a function which is nonnegative and integrable. Then, for integral operators (9) and (10), the following inequalities hold: where for , for and .

Proof. Multiplying both sides of (21) with and integrating over , we have

Putting in (36), we get

By using (10) and given condition , the first inequality of (35) is obtained.

Now, multiplying both sides of (22) with and integrating over , we have

From the above inequality, the second inequality of (35) can be achieved.

The following remark establishes the connection with already published results.

Remark 14. (i)If we take in (35), then Fejér-Hadamard inequality for the exponentially -convex function of second kind is obtained(ii)If we take in (35), then Fejér-Hadamard inequality for the exponentially convex function is obtained(iii)If we take , , and in (35), then [35] (Theorem 2.2) is obtained(iv)If we take , , and in (35), then [35] (Theorem 3.2) is obtained(v)If we take , , , and in (35), then [27] (Theorem 2.2) is obtained(vi)If we take , , , and in (35), then [33] (Theorem 4) is obtained(vii)If we take , , , , and in (35), then classical Fejér-Hadamard inequality is obtained

In the following, we give another generalized fractional version of the Fejér-Hadamard inequality.

Theorem 15. Let , be the real-valued functions. If be a integrable, exponentially -convex and and be a differentiable and strictly increasing. Also, let be a function which is nonnegative and integrable. Then, for integral operators (9) and (10), the following inequalities hold: where for , for and .

Proof. Multiplying both sides of (30) with and integrating over , we have Putting in (40), we get By using (10) and the given condition , the first inequality of (39) is obtained.
Now, multiplying both sides of (31) with and integrating over , we have From the above inequality, the second inequality of (39) can be achieved.

Remark 16. (i)If we take in (39), then Fejér-Hadamard inequality for the exponentially -convex function of second kind is obtained(ii)If we take in (39), then Fejér-Hadamard inequality for the exponentially convex function is obtained

4. Concluding Remarks

We have established two generalized fractional versions of the Hadamard inequality. Also, two generalized fractional versions of the Fejér-Hadamard inequality are proved. For proving these fractional inequalities, we have utilized exponentially -convex functions as well as generalized integral operators containing Mittag-Leffler functions in their kernels. The presented results hold for various kinds of convexities and well-known fractional integral operators given in Remarks 6 and 8. Therefore, the reader can get a lot of desired fractional Hadamard and fractional Fejér-Hadamard inequalities from this paper.

Data Availability

All data are included within this paper.

Conflicts of Interest

The authors do not have any competing interests.

Authors’ Contributions

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

Acknowledgments

This work was sponsored in part by Chongqing University of Arts and Sciences (Z2018SC09) and the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).

References

G. Farid, “Some Riemann-Liouville fractional integral inequalities for convex functions,” Journal of Analysis, vol. 27, no. 4, pp. 1095–1102, 2019.
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
S. Mehmood and G. Farid, “-convex functions associated with bounds of -fractional integrals,” Advances in Inequalities and Applications, vol. 2020, 2019.
View at: Google Scholar
S. Mehmood and G. Farid, “Fractional integrals inequalities for exponentially -convex functions,” Open Journal of Mathematical Sciences, vol. 4, no. 1, pp. 78–85, 2020.
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, no. 1, 2019.
View at: Publisher Site | Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
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: Publisher Site | Google Scholar
G. Farid, “A unified integral operator and further its consequences,” Open Journal of Mathematical Analysis, vol. 4, no. 1, pp. 1–7, 2020.
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
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
H. M. Srivastava and Z. 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
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
C. P. Niculescu and L. E. Persson, Convex Functions and Their Applications, a Comtemporary Approach, CMS Books in Mathematics, vol. 23, Springer Verlag, New York, 2006.
View at: Publisher Site
J. Pecaric, F. Proschan, and Y. L. Tong, “Convex Functions,” in Partial Orderings and Statistical Applications, Academics Press, New York, 1973.
View at: Google Scholar
A. W. Roberts and D. E. Varberg, Convex Functions, Academics Press, New York, USA, 1973.
X. Qiang, G. Farid, J. Pecaric, and S. B. Akbar, “Generalized fractional integral inequalities for exponentially (s,m)$(s,m)$-convex functions,” Journal of Inequalities and Applications, vol. 2020, no. 1, 2020.
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, 2019.
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
N. Eftekhari, “Some remarks on (s,m)-convexity in the second sense,” Journal of Mathematical Inequalities, vol. 8, no. 3, pp. 489–495, 2007.
View at: Publisher Site | Google Scholar
H. Hudzik and L. Maligranda, “Some remarks ons-convex functions,” Aequationes Mathematicae, vol. 48, no. 1, pp. 100–111, 1994.
View at: Publisher Site | Google Scholar
G. H. Toader, “Some generaliztion of convexity”.
View at: Google Scholar
L. Fejér, “Überdie fourierreihen II,” Math Naturwiss Anz Ungar Akad Wiss, vol. 24, pp. 369–390, 1906.
View at: Google Scholar
G. Abbas and G. Farid, “Hadamard and Fejér-Hadamard type inequalities for harmonically convex functions via generalized fractional integrals,” Journal of Analysis, vol. 25, no. 1, pp. 107–119, 2017.
View at: Publisher Site | Google Scholar
F. Chen, “On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity,” Chinese Journal of Mathematics, vol. 2014, 7 pages, 2014.
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
S. S. Dragomir and I. Gomm, “Some Hermite-Hadamard type inequalities for functions whose exponentials are convex,” Studia Universitatis Babeș-Bolyai Mathematica, vol. 60, no. 4, pp. 527–534, 2015.
View at: Google Scholar
G. Farid, “Hadamard and Fejér-Hadamard inequalities for generalized fractional integral involving special functions,” Konuralp Journal of Mathematics, vol. 4, no. 1, pp. 108–113, 2016.
View at: Google Scholar
G. Farid, “A treatment of the Hadamard inequality due to -convexity via generalized fractional integrals,” Journal of Fractional Calculus and Applications, vol. 9, no. 1, pp. 8–14, 2018.
View at: Google Scholar
G. Farid and G. Abbas, “Generalizations of some fractional integral inequalities for -convex functions via generalized Mittag-Leffler function,” Studia Universitatis Babeș-Bolyai Mathematica, vol. 63, no. 1, pp. 23–35, 2018.
View at: Publisher Site | Google Scholar
G. Farid, “On Hadamard-type inequalities for m-convex functions via Riemann-Liouville fractional integrals,” Studia Universitatis Babes-Bolyai Matematica, vol. 62, no. 2, pp. 141–150, 2017.
View at: Publisher Site | Google Scholar
G. Farid, A. U. Rehman, and S. Mehmood, “Hadamard and Fejer-Hadamard type integral inequalities for harmonically convex functions via an extended generalized Mittag-Leffler function,” The Journal of Mathematics and Computer Science, vol. 8, no. 5, pp. 630–643, 2018.
View at: Publisher Site | 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, 2018.
View at: Publisher Site | Google Scholar
I. Işcan, “Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals,” Studia Universitatis Babeș-Bolyai Mathematica, vol. 60, no. 3, pp. 355–366, 2015.
View at: Google Scholar
S. M. Kang, G. Farid, W. Nazeer, and S. Mehmood, “(h − m) $(h-m)$ -convex functions and associated fractional Hadamard and Fejér-Hadamard inequalities via an extended generalized Mittag-Leffler function,” Journal of Inequalities and Applications, vol. 2019, no. 1, 2019.
View at: Publisher Site | Google Scholar
S. M. Kang, G. Farid, W. Nazeer, and B. Tariq, “Hadamard and Fejér-Hadamard inequalities for extended generalized fractional integrals involving special functions,” Journal of Inequalities and Applications, vol. 2018, no. 1, 2018.
View at: Google Scholar
M. Z. Sarikaya, E. Set, H. Yaldiz, and N. Basak, “Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities,” Mathematical and Computer Modelling, vol. 57, no. 9-10, pp. 2403–2407, 2013.
View at: Publisher Site | Google Scholar
M. Z. Sarikaya and H. Yildirim, “On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals,” Miskolc Mathematical Notes, vol. 17, no. 2, pp. 1049–1059, 2017.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Shuya Guo 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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