New Fractional Mercer–Ostrowski Type Inequalities with Respect to Monotone Function

Butt, Saad Ihsan; Nosheen, Ammara; Nasir, Jamshed; Khan, Khuram Ali; Matendo Mabela, Rostin

doi:https://doi.org/10.1155/2022/7067543

Mathematical Problems in Engineering

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Volume 2022 | Article ID 7067543 | https://doi.org/10.1155/2022/7067543

New Fractional Mercer–Ostrowski Type Inequalities with Respect to Monotone Function

Saad Ihsan Butt,¹Ammara Nosheen,²Jamshed Nasir,³Khuram Ali Khan,²and Rostin Matendo Mabela⁴

Academic Editor: Muhammad Irfan

Received22 Dec 2021

Accepted09 Apr 2022

Published18 May 2022

Abstract

This research focuses on Ostrowski type inequality in the form of classical Mercer inequality via -Riemann–Liouville fractional integral (F-I) operators. Using the -Riemann–Liouville F-I operator, we first develop and demonstrate a new generalized lemma for differentiable functions. Based on this lemma, we derive some fractional Mercer–Ostrowski type inequalities by using the convexity theory. These new findings extend and recapture previous published results. Finally, we presented applications of our work via the known special functions of real numbers such as q-digamma functions and Bessel function.

1. Introduction and Preliminaries

The well-known Ostrowski inequality, developed in 1938, established the following helpful and noteworthy integral inequality (see [1], page 468).

Suppose a mapping is continuous on and differentiable on . If , for all , then, the following inequality holds:

This finding is known as the Ostrowski inequality in the literature. Some generalizations, variations, and extensions of the Ostrowski inequality have been proposed in light of current findings and their related generalizations, variants, and extensions (see [2–4]).

This inequality yields an upper bound for the approximation of the integral average by the value of at the point .

In recent years, because of the widespread interest in the theory of inequalities, the theory of convex functions is now at the center of many studies. Convex functions are the topic of research in a number of disciplines due to their applicability in inequality theory [5–8] and defined as:

Definition 1. [5] A mapping is called to be convex on , ifholds for every and .
Kian and Moslehian used the Jensen–Mercer inequality and demonstrated the Hermite–Hadamard Mercer inequality in [9] as:where is convex function on .
The famous Jensen inequality, (see [5], Ch. 1) in the literature states that, if is convex function on the interval , thenholds for all and all .
Jensen’s inequality was modified by Mercer (see [10]) aswhere is a convex function on holds for all andJensen and Hermite–Hadamard type inequalities are the most dynamic inequalities pertaining convex functions. Jensen and its related inequalities are well-known and significant inequalities in mathematical analysis due to its diverse applications and useability in applied and information sciences. Some recent discoveries can be found in [11, 12].
Jensen-type Mercer’s inequality is an effective inequality since it provides additional information with specific boundary constraints. The study of generalizations and improvements of Mercer’s variants of Hermite–Hadamard type inequalities considering the variety of fractional integral (F-I) operators have been of great interest for researchers in recent years, as evidenced by a large amount of research on it (see [13–15]).
The fractional calculus has been extensively studied by many researchers from the last few decades to generalize, improve, and extend several classic inequalities in order to obtain new variants in different dimensions. There are not only global derivatives in so called fractional calculus (for example: Riemann–Liouville and Caputo), but also local fractional derivatives (Khalil and Almeida, among others) (see [16–18]).
Yue [19], in 2013, discovered new Ostrowski inequalities for fractional integral (F-I) operators along with its associated fractional inequalities. Later in 2014, Aljinović [20] first developed Montgomery identity for fractional integrals of one function with respect to another function and then derived generalized fractional Ostrowski inequality from it. In the same article, he also presented the associated Ostrowski fractional inequalities for fractional integrals of functions with first derivatives in spaces and computed sharp bounds. In the same year, Yildirim and Kirtay [21] used the generalized Riemann–Liouville F-I to establish new variants for Ostrowski inequalities. Some recent development about weighted Ostrowski fractional inequalities can be observed in [22].
Vanterler da Costa Sousa and Capelas de Oliveira in [23] recently introduced the -Hilfer fractional derivative with respect to another function. Also, they investigate some Gronwall inequalities and Cauchy–type problem using the newly introduced -Hilfer operator.

Definition 2. ([24], p.3) Let be finite or infinite interval in and . Also, let be positive strictly increasing function possessing continuous derivative on . Then, left- and right-sided -Riemann–Liouville F-I of a function with respect to another function on can be given asand

Remark 1. F-I operators elaborated in (7) and (8) yield several known F-I operators corresponding to various suitable selections of function (see [25]), that are independently introduced by several authors with related results.(i)By taking as an identity function in (7) and (8), we get Riemann–Liouville F-I operators [24].(ii)For , in (7) and (8) produce Katugampola F-I operators defined by Chen and Katugampola in [26].(iii)For , in (7) and (8) produce generalized conformable F-I operators defined by Khan and Khan in [27].(iv)For , , in (7), and , , in (8), one can get conformable F-I operators presented by Jarad et al. in [28].The main good articles about Hermite–Hadamard inequalities involving -Riemann–Liouville F-I operators are in references [29–31]. Some recent results about Hermite–Jensen–Mercer inequalities for -Riemann–Liouville F-I operators can be seen in [14, 15].
The striking motive of this study is to develop generalized fractional equality for -Riemann–Liouville F-I operators, which has a unique place among fractional integral operators, and to use this identity to generate some new Mercer–Ostrowski type inequalities for convex functions. The study also included applications of the findings, taking into account several specific circumstances of the primary conclusions.

2. New Mercer–Ostrowski Type Inequalities

Throughout this portion, Mercer–Ostrowski inequalities for the -Riemann–Liouville F-I operators are obtained for differentiable functions on . As a result, we present a novel identity pertaining -Riemann–Liouville F-I operators, that will serve as an auxiliary equality to produce subsequent inequalities.

Lemma 1. Consider be a differentiable function and , with . If is a strictly increasing, positive monotone function on with continuous derivative on , then, for all and , the following identity holdswhere

Proof. Let us start with and similarly, we getIt follows from (11) and (13) thatBy simplifying, we get the required result.

Remark 2. Placing identity function in (9), then, we get Lemma 2.1 given in [32].

Remark 3. If we set , , and in (9), it reduces to Lemma 2 in [33].

Remark 4. Setting , , , and in (9), it recaptures Lemma 1 proved in [2].

Theorem 1. Under the assumptions of Lemma 1, if is convex function on , then for all , the following inequality is valid.

Proof. By means of (9),Change of variables and and then into the resulting equality, we getSince is convex function on , we obtain

Remark 5. If we set in Theorem 1, one can get above inequality for Riemann–Liouville F-I operators given in Theorem 2.1 [32].

Remark 6. If we set , , and in Theorem 1, it reduces to Theorem 7 in [33] that yields the same results with .

Corollary 1. If we set , , and with in Theorem 1, we get the following inequality

Corollary 2. If we set and in Theorem 1, we get the following Mercer–Ostrowski inequality:

Corollary 3. The following Mercer–Ostrowski inequality can be found in Theorem 1 with

Proof. The result can be obtained by using and

Remark 7. If we set , and in Corollary 3, it reduces to Corollary 1 in [33].

Remark 8. If we set , and and in Corollary 3, it reduces to Theorem 2 in [2] that yields the same result with .

Theorem 2. We assume that all the conditions of Lemma 1 hold. If is convex function on , then, for all , the following inequalityholds, where are conjugate exponents.

Proof. Applying classical Hölder integral inequality and the convexity of on the right side of (9), we getThat finish the proof.

Remark 9. If we set in Theorem 2, it reduces to Theorem 2.2 in [32].

Remark 10. If we set , , and in Theorem 2, it reduces to Theorem 8 in [33] that yields the same results with .

Corollary 4. If we set , , and with in Theorem 2, we have the following inequality:

Corollary 5. If we set and in Theorem 2, we lead to following Mercer–Ostrowski inequality:

Corollary 6. Let the function in Theorem 2 is assumed to be bounded that is , then the following result holds:

Proof. The result can be demonstrated by using and

Remark 11. If we set , , and in Corollary 6, it reduces to Corollary 2 in [33].

Remark 12. If we set , , and in Corollary 6, it reduces to Theorem 3 in [2] that yields the same result with .

Theorem 3. We assume that all the conditions of Lemma 1 hold. If is convex function on , then for all , the following inequalityis valid.

Proof. Applying power-mean integral inequality and the convexity of on the right side of (9), we getWhich ends the proof.

Remark 13. If we set in Theorem 3, it reduces to Theorem 2.3 in [32].

Remark 14. If we set , , and in Theorem 3, it reduces to Theorem 9 in [33] that yields the same results with .

Corollary 7. If we set , , and with in Theorem 3, we have the following inequality:

Corollary 8. If we choose and in Theorem 3, we have the following Mercer–Ostrowski inequality:

Corollary 9. Assuming that , in Theorem 3, the following Mercer–Ostrowski inequality holds:

Proof. Under the assumed conditions, we have and . Thus, inequality (27) in Theorem 3 leads to inequality (31).

Remark 15. If we set , , and in Corollary 9, it reduces to Corollary 3 in [33].

Remark 16. If we set , , , and in Corollary 9, it reduces to Theorem 4 in [2] that yields the same result with .

Theorem 4. We assume that all the assumptions of Lemma 1 holds. If is convex function on for all , the following inequalityholds where are conjugate exponents.

Proof. From (9), we obtainUtilizing Young’s inequality asBy the convexity of , we haveand the proof is done.

Remark 17. In Theorem 4, putting , then, one can get Mercer–Ostrowski inequality pertaining Riemann–Liouville F-I operators given in [32].

Corollary 10. If we choose and in Theorem 4, the following Mercer–Ostrowski inequality holds:

Corollary 11. The following Mercer–Ostrowski inequality can be obtained from Theorem 4 by assuming :

Theorem 5. We assume that all the assumptions of Lemma 1 hold. If is concave function on , for all , the following inequality holds:where are conjugate exponents.

Proof. From (9), by using the Hölder’s inequality, we obtainSince is concave mapping, therefore from (3), we obtainandBy placing the inequalities (40) and (41) in (39), leads to (38).

Remark 18. If we set in Theorem 5, it reduces to Theorem 2.5 in [32].

Remark 19. If we set , , , and in Theorem 5, it reduces to Theorem 5 in [2] that yields the same result with .

Corollary 12. If we choose and in Theorem 5, we deduce the Mercer–Ostrowski inequality as

3. Some Applications

3.1. Applications to Means

For two real numbers , consider the following two important means:

The arithmetic mean:

The generalized logarithmic-mean:

Proposition 1. Suppose then, we have the following inequality

Proof. The result can be obtained immediately by taking into account Corollary 8 along with the convex function .

Proposition 2. Suppose then, we have the following inequality:

Proof. The proof is the direct consequence of Corollary 10 by considering the convex function .

3.2. q-Digamma Function

The -digamma function, which is described as the logarithmic derivative of the -gamma function, is an essential function related to the q-gamma function (see [34]) given as

For and , q-digamma function can be given as

Proposition 3. Assume that are the real numbers such that , , , and . Then, the following inequality is valid:

Proof. The statement can be obtained by using Corollary 8 by considering . Since, is a completely monotone function on for all , consequently, is convex on the same interval , (see [34]).

3.3. Bounds Involving Modified Bessel Function

We know the first type of modified Bessel function , which has the series interpretation (see [35], p.77)where and , while the second kind modified Bessel function (see [35], p.78) is usually defined as

Consider the function defined by

The first- and second-order derivative formula of is given as [35]:

Proposition 4. Suppose that and . Then, we have

Proof. Substituting the mapping to the inequality in Corollary 10. Note that all assumptions of Corollary 10 are satisfied (see [34]). Therefore, using the identities (53) and (54) gives required result.

4. Conclusions

The objective of this study is to introduce the idea of new generalized fractional variants of Ostrowski inequality by employing Jensen–Mercer inequality for differentiable convex functions. The obtained results are intersting and generalized in a sense that by substituting identity function and special value of , we get connected to previously established results in the literature. Also, one can get variety of fractional Mercer–Ostrowski inequalities for different F-I operators by considering particular values of function as mentioned in Remark 1. In addition, another motivating aspect of the study is that we try to give applications of means, q-digamma function, and Bessel function for Mercer–Ostrowski inequality in the similar passion as considered for Hermite–Hadamard type inequalities given in [8, 30]. Based on this study, researchers may contribute to the development of such results for twice differentiable functions.

Data Availability

No data are available.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

D. S. Mitrinovic, J. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Springer Science and Business Media, Berlin, Germany, vol. 53, 1991.
M. Alomari, M. Darus, S. S. Dragomir, and P. Cerone, “Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1071–1076, 2010.
View at: Publisher Site | Google Scholar
S. S. Dragomir and T. M. Rassias, Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic, Dordrecht, Netherlands, 2002.
A. Ekinci, M. E. Ozdemir, and E. Set, “New integral inequalities of Ostrowski type for quasi-convex functions with applications,” Turkish Journal of Science, vol. 5, no. 3, pp. 290–304, 2021.
View at: Google Scholar
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, “Classical and New Inequalities in Analysis,” Mathematics and its Applications (East European Series), Kluwer Academic Publishers Group, Dordrecht, Netherlands, vol. 61, 1993.
View at: Google Scholar
Y. Qin, Integral and Discrete Inequalities and Their Applications, Springer International Publishing Switzerland, Birkhauser Basel, 2016.
P. Agarwal, S. S. Dragomir, M. Jleli, and B. Samet, Advances in Mathematical Inequalities and Applications, Springer, Singapore, 2018.
K. Mehrez and P. Agarwal, “New Hermite-Hadamard type integral inequalities for convex functions and their applications,” Journal of Computational and Applied Mathematics, vol. 350, pp. 274–285, 2019.
View at: Publisher Site | Google Scholar
M. Kian and M. Moslehian, “Refinements of the operator Jensen-Mercer inequality,” The Electronic Journal of Linear Algebra, vol. 26, 2013.
View at: Publisher Site | Google Scholar
A. M. Mercer, “A variant of Jensen’s inequality,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, Article ID 73, 2003.
View at: Google Scholar
S. Khan, M. Adil Khan, S. I. Butt, and Y.-M. Chu, “A new bound for the Jensen gap pertaining twice differentiable functions with applications,” Advances in Difference Equations, vol. 2020, no. 1, p. 333, 2020.
View at: Publisher Site | Google Scholar
S. I. Butt, M. Klaričić Bakula, Đ. Pečarić, and J. Pečarić, “Jensen‐grüss inequality and its applications for the zipf‐mandelbrot law,” Mathematical Methods in the Applied Sciences, vol. 44, no. 2, pp. 1664–1673, 2021.
View at: Publisher Site | Google Scholar
H.-H. Chu, S. Rashid, Z. Hammouch, and Y. M. Chu, “New fractional estimates for Hermite-Hadamard-Mercer’s type inequalities,” Alexandria Engineering Journal, vol. 59, no. 5, pp. 3079–3089, 2020.
View at: Publisher Site | Google Scholar
S. I. Butt, A. Kashuri, M. Umar, A. Aslam, and W. Gao, “Hermite-Jensen-Mercer type inequalities via -Riemann-Liouville k-fractional integrals,” AIMS Mathematics, vol. 5, no. 5, Article ID 51935220, 2020.
View at: Publisher Site | Google Scholar
S. I. Butt, M. Umar, K. A. Khan, A. Kashuri, and H. Emadifar, “Fractional Hermite-Jensen-Mercer Integral Inequalities with Respect to Another Function and Application,” Complexity, vol. 2021, Article ID 9260828, 30 pages, 2021.
View at: Publisher Site | Google Scholar
R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, “A new definition of fractional derivative,” Journal of Computational and Applied Mathematics, vol. 264, pp. 65–70, 2014.
View at: Publisher Site | Google Scholar
P. O. Mohammed, H. Aydi, A. Kashuri, Y. S. Hamed, and K. M. Abualnaja, “Midpoint inequalities in fractional calculus defined using positive weighted symmetry function kernels,” Symmetry, vol. 13, p. 550, 2021.
View at: Google Scholar
H. Aydi, M. Jleli, and B. Samet, “On positive solutions for a fractional thermostat model with a convex-concave source term via $$\psi $$-Caputo fractional derivative,” Mediterranean Journal of Mathematics, vol. 17, no. 1, p. 16, 2020.
View at: Publisher Site | Google Scholar
H. Yue, “Ostrowski inequality for fractional integrals and related fractional inequalities,” Transylvanian Journal of Mathematics and Mechanics, vol. 5, pp. 85–89, 2013.
View at: Google Scholar
A. A. Aljinović, “Montgomery identity and Ostrowski type inequalities for Riemann–Liouville fractional integral,” Journal of Mathematics, vol. 2014, Article ID 503195, 6 pages, 2014.
View at: Publisher Site | Google Scholar
H. Yildirim and Z. Kirtay, “Ostrowski inequality for generalized fractional integral and related inequalities,” Malaya Journal of Matematik, vol. 2, pp. 322–329, 2014.
View at: Google Scholar
A. Kashuri, B. Meftah, P. O. Mohammed et al., “Fractional weighted ostrowski-type inequalities and their applications,” Symmetry, vol. 13, no. 6, p. 968, 2021.
View at: Publisher Site | Google Scholar
J. Vanterler da Costa Sousa and E. Capelas de Oliveira, “A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator,” Differential Equations & Applications, vol. 11, no. 1, pp. 87–106, 2019.
View at: Publisher Site | Google Scholar
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, NY, USA, 2006.
G. Farid, “Study of a generalized Riemann-Liouville fractional integral via convex functions,” Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, vol. 69, no. 1, pp. 37–48, 2020.
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
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
F. Jarad, E. Ugurlu, T. Abdeljawad, and D. Baleanu, “On a new class of fractional operators,” Advances in Difference Equations, vol. 2017, no. 1, pp. 1–16, 2017.
View at: Publisher Site | Google Scholar
K. Liu, J. Wang, and D. O’regan, “On the Hermite-Hadamard type inequality for -Riemann-Liouville fractional integrals via convex functions,” Journal of Inequalities and Applications, vol. 2019, Article ID 27, 2019.
View at: Publisher Site | Google Scholar
P. O. Mohammed, “Hermite‐Hadamard inequalities for Riemann‐Liouville fractional integrals of a convex function with respect to a monotone function,” Mathematical Methods in the Applied Sciences, vol. 44, no. 3, pp. 2314–2324, 2021.
View at: Publisher Site | Google Scholar
S. S. Dragomir, “Hermite‐Hadamard type inequalities for generalized Riemann‐Liouville fractional integrals of h ‐convex functions,” Mathematical Methods in the Applied Sciences, vol. 44, no. 3, pp. 2364–2380, 2021.
View at: Publisher Site | Google Scholar
J. Nasir, S. I. Butt, S. Qaisar, E. Set, and A. O. Akdemir, “Some Mercer–Ostrowski type inequalities for differentiable convex functions via fractional integral operators”.
View at: Google Scholar
E. Set, “New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals,” Computers & Mathematics with Applications, vol. 63, no. 7, pp. 1147–1154, 2012.
View at: Publisher Site | Google Scholar
S. Jain, K. Mehrez, D. Baleanu, and P. Agarwal, “Certain hermite-hadamard inequalities for logarithmically convex functions with applications,” Mathematics, vol. 7, no. 2, p. 163, 2019.
View at: Publisher Site | Google Scholar
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 1995.

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