1. Introduction
Computational and Fractional Analysis are nowadays more and more in the center of mathematics and of other related sciences either by themselves because of their rapid development, which is based on very old foundations, or because they cover a great variety of applications in the real world. In current years, fractional calculus
applied in many phenomena in applied sciences, fluid mechanics, physics and other biology can be described as very effective using mathematical tools of
. The fractional derivatives have occurred in many applied sciences equations such as reaction and diffusion processes, system identification, velocity signal analysis, relaxation of damping behaviour fabrics and creeping of polymer composites [
1,
2,
3,
4].
Definition 1 ([
5])
. Let I be an interval of real numbers. A function is said to be convex, if for all and all , we have The concept of convex functions has been also generalized in diverse manners. One of them is the quasi-convex function defined as follows:
Definition 2 ([
6])
. A function is said to be quasi-convex, ifholds for all and . In 1938 Ostrowski ([
7]) proved the following Ostrowski inequality:
Theorem 1 ([
7])
. Let φ be a differentiable function defined on the finite interval , whose derivative is integrable and bounded over then Several generalizations, improvement, and variants of such type of inequality have been obtained, we refer readers to [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21] and references therein.
The following notations will be used in the sequel. We denote, respectively the interior of I and the set of all integrable functions on
In [
22], Alomari et al. gave the following midpoint type inequalities for differentiable quasi-convex.
Theorem 2 ([
22])
. Let be a differentiable mapping on such that , where with . If is quasi-convex on , then the following inequality holds: Theorem 3 ([
22])
. Let be a differentiable mapping on such that , where with . If is quasi-convex on , with , then the following inequality holds: Theorem 4 ([
22])
. Let be a differentiable mapping on such that , where with . If is quasi-convex on , , then the following inequality holds: Alomari and Darus in [
23] obtained the Ostrowski-type inequalities for differentiable quasi-convex functions:
Theorem 5 ([
23])
. Let be a differentiable mapping on such that , where with . If is quasi-convex on , then the following inequality holds: Theorem 6 ([
23])
. Let be a differentiable mapping on such that , where with . If is quasi-convex on , with , then the following inequality holds: Theorem 7 ([
23])
. Let be a differentiable mapping on such that , where with . If is quasi-convex on , , then the following inequality holds: Fractional calculus has been widely studied by many researchers over the past decades. In particular, to generalize classic inequalities. Among the best known and use of these fractional integral operators we recall that of Riemann–Liouville.
Definition 3 ([
24])
. Let . The Riemann–Liouville integrals and of order with are defined byrespectively, where , is the Gamma function and . In so-called fractional calculus, there are not only global derivatives (for example, Riemann–Liouville and Caputo), but also local fractional derivatives (Khalil, Almeida, among others), see [
25,
26]. Regarding some papers dealing with fractional integral inequalities via different types of fractional integral operators, we refer readers to [
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43].
In this paper, we establish a new identity and then apply it to derive new weighted Ostrowski-type inequalities for quasi-convex functions. Further results for functions with a bounded first derivative will be given. In order to illustrate the efficiency of our main results, some applications to special means will be obtain.
2. Main Results
For brevity, we will used in the sequel In order to find our main results, we need to prove the following lemma.
Lemma 1. Letbe differentiable function on,
,
anda continuous function. If, thenwhereand Proof. We integrate by parts and change the variable
to obtain
Now, multiplying (
4) by
, we get
Multiplying (
6) by
, we obtain
By taking the difference between (
5) and (
7), we get
which is the desired result. □
Theorem 8. Consider a differentiable functionwithwhere, and letbe continuous function. Ifis quasi-convex, then Proof. From Lemma 1, properties of modulus and quasi-convexity of
, we have
The proof is completed. □
Corollary 1. Considering in Theorem 8, we obtain Corollary 2. Choosing in Theorem 8, we get Moreover, if we take , we obtain Corollary 3. Let in Theorem 8, then Moreover, for , we get Remark 1. In Corollary 3, if we choose , we obtain ([23], Theorem 2). Moreover, if we take , we get ([22], Theorem 7). Theorem 9. Consider a differentiable function with where , and let be continuous function. If is quasi-convex, and , then Proof. Applying properties of modulus, Lemma 1, Hölder’s inequality, and quasi-convexity of
, we obtain
The proof is completed. □
Corollary 4. Let in Theorem 9, then Corollary 5. Choosing in Theorem 9, we get Moreover, if we take , we obtain Corollary 6. For in Theorem 9, we have Moreover, considering , we get Corollary 7. In Corollary 6, if we choose , we obtain For , we get ([22], Theorem 8). Theorem 10. Consider a differentiable function with where , and let be continuous function. If is quasi-convex, where , then Proof. By properties of modulus, applying Lemma 1, power mean inequality and quasi-convexity of
, we obtain
The proof is completed. □
Corollary 8. For in Theorem 10, we get Corollary 9. Choosing in Theorem 10, we have Moreover, if we take , we obtain Corollary 10. Considering in Theorem 10, we get Moreover, for , we have Remark 2. Choosing in Corollary 10, we obtain ([23], Theorem 4). Moreover, for , we get ([22], Theorem 9). 3. Further Results
In this section, we will prove the following results.
Theorem 11. Consider a differentiable function with , , and let be continuous function. If there exist constants such that for all , thenwhereand are defined as in (2) and (3), respectively. Proof. From (
9), we have
where
is defined as in (
8). Applying absolute value on both sides of (
10) and using the fact that
for all
, we obtain
The proof is completed. □
Corollary 11. For in Theorem 11, we have Corollary 12. Putting in Theorem 11, we get Moreover, if we take , we obtain Corollary 13. Choosing in Theorem 11, we have Moreover, for , we get Corollary 14. Let in Corollary 13, then Before giving our next result, we recall that a function
is
r-
H-Hölder (Hölder condition, see [
44]), if
holds for all
, for some
and
.
Theorem 12. Consider a differentiable function with , , and let be continuous function. If satisfies Hölder condition for some and , thenwhereand are defined as in (2) and (3), respectively. Proof. Applying Lemma 1, we have
From (
12), we get
where
is defined as in (
11). Applying absolute value on both sides of (
13) and
r-
H-Hölder property of
, we obtain
The proof is completed. □
Corollary 15. Letting in Theorem 12, we have Corollary 16. Choosing in Theorem 12, we get Moreover, if we take , we obtain Corollary 17. For in Theorem 12, we have Moreover, for , we get Corollary 18. In Corollary 17, if we choose , we obtain Moreover, if we take , we have Corollary 19. Let be a L–Lipschitzian function ( and ) in Theorem 12, thenwhere is defined by (11). Corollary 20. For in Corollary 19, we get Corollary 21. In Corollary 19, if we choose , we obtain Moreover, if we take , we have Corollary 22. Choosing in Corollary 19, we get Moreover, for , we obtain Corollary 23. For in Corollary 22, we have Moreover, for , we get 4. Applications
We shall consider the following special means for different positive real numbers a and where :
Arithmetic mean: .
p–Logarithmic mean: ,.
Let recall from [
23] the following quasi-convex functions,
and
for all
respectively.
Using
Section 2, we are in position to prove the following results regarding above special means.
Proposition 1. Let , , and then Proof. Taking for and in Corollary 3, we get the desired result. □
Proposition 2. Let , , and then Proof. Choosing for and in Corollary 3, we obtain the desired result. □
Proposition 3. Let , , and then Proof. Taking for and in Corollary 6, we get the desired result. □
Proposition 4. Let , , and then Proof. Choosing for and in Corollary 6, we obtain the desired result. □
5. Conclusions
The main results and future research of the article can be summarized as follows:
A new identity regarding fractional weighted Ostrowski-type is established.
New fractional weighted Ostrowski-type inequalities for quasi-convex functions using the above identity are deduced.
Several further results for function with a bounded first derivative are given.
Some applications to special means are obtained.
The efficiency of our results is shown.
As future research, from our results, interested reader can find several new interesting inequalities from many areas of pure and applied sciences. Moreover, they can derive (using our technique) applications to special means for different quasi-convex functions.
Author Contributions
Conceptualization, A.K., B.M., P.O.M., T.A.; methodology, B.M., P.O.M., A.A.L., Y.S.H.; software, P.O.M., Y.S.H., A.K.; validation, P.O.M., Y.S.H., B.A.; formal analysis, P.O.M., Y.S.H., T.A.; investigation, P.O.M.; resources, P.O.M., A.K., A.A.L.; data curation, P.O.M., Y.S.H.; writing—original draft preparation, B.M., Y.S.H.; writing—review and editing, Y.S.H., P.O.M., B.A.; visualization, Y.S.H.; supervision, P.O.M., Y.S.H., T.A., A.K. All authors have read and agreed to the final version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
This work was supported by the Taif University Researchers Supporting Project (No. TURSP-2020/155), Taif University, Taif, Saudi Arabia, and the authors Bahaaeldin Abdalla and Thabet Abdeljawad would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Hilfer, R. Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000; Volume 35, pp. 87–130. [Google Scholar]
- Cesarano, C.; Pierpaolo, N.; Paolo, E.R. Pseudo-Lucas functions of fractional degree and applications. Axioms 2021, 10, 51. [Google Scholar] [CrossRef]
- Sabatier, J.; Agrawal, O.P.; Machado, J.T. Advances in Fractional Calculus; Springer: Dordrecht, The Netherlands, 2007; Volume 4. [Google Scholar]
- Al-luhaibi, M.S. An analytical treatment to fractional Fornberg-Whitham equation. Math. Sci. 2017, 11, 1–6. [Google Scholar] [CrossRef] [Green Version]
- Pečarić, J.E.; Proschan, F.; Tong, Y.L. Convex functions, partial orderings, and statistical applications. In Mathematics in Science and Engineering; Academic Press, Inc.: Boston, MA, USA, 1992; Volume 187. [Google Scholar]
- Ion, D.A. Some estimates on the Hermite–Hadamard inequality through quasi-convex functions. An. Univ. Craiova Ser. Mat. Inform. 2007, 34, 83–88. [Google Scholar]
- Ostrowski, A. Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert. (German) Comment. Math. Helv. 1937, 10, 226–227. [Google Scholar] [CrossRef]
- Anastassiou, G.A. Ostrowski-type inequalities. Proc. Amer. Math. Soc. 1995, 123, 3775–3781. [Google Scholar] [CrossRef]
- Awan, M.U.; Akhtar, N.; Kashuri, A.; Noor, M.A.; Chu, Y.M. 2D approximately reciprocal ρ–convex functions and associated integral inequalities. AIMS Math. 2020, 5, 4662–4680. [Google Scholar] [CrossRef]
- Dragomir, S.S.; Wang, S. An inequality of Ostrowski–Grüss type and its applications to the estimation of error bounds for some special means and for some numerical quadrature rules. Comput. Math. Appl. 1997, 33, 15–20. [Google Scholar] [CrossRef] [Green Version]
- Kashuri, A.; Liko, R. Generalizations of Hermite–Hadamard and Ostrowski-type inequalities for MTm–preinvex functions. Proyecciones 2017, 36, 45–80. [Google Scholar] [CrossRef] [Green Version]
- Kashuri, A.; Liko, R. Ostrowski-type fractional integral operators for generalized (r, s, m, φ)–preinvex functions. Appl. Appl. Math. 2017, 12, 1017–1035. [Google Scholar] [CrossRef] [Green Version]
- Kashuri, A.; Du, T.S.; Liko, R. On some new integral inequalities concerning twice differentiable generalized relative semi–(m, h)–preinvex mappings. Stud. Univ. Babeş-Bolyai Math. 2019, 64, 43–61. [Google Scholar] [CrossRef] [Green Version]
- Kunt, M.; Kashuri, A.; Du, T.S.; Baidar, A.W. Quantum Montgomery identity and quantum estimates of Ostrowski-type inequalities. AIMS Math. 2020, 5, 5439–5457. [Google Scholar] [CrossRef]
- Meftah, B. Ostrowski inequality for functions whose first derivatives are s–preinvex in the second sense. Khayyam J. Math. 2017, 3, 61–80. [Google Scholar]
- Meftah, B. Some new Ostrowski’s inequalities for n–times differentiable mappings which are quasi-convex. Facta Univ. Ser. Math. Inform. 2017, 32, 319–327. [Google Scholar]
- Meftah, B. Some new Ostrowski’s inequalities for functions whose nth derivatives are logarithmically convex. Ann. Math. Sil. 2017, 32, 275–284. [Google Scholar] [CrossRef] [Green Version]
- Meftah, B. Some Ostrowski’s inequalities for functions whose nth derivatives are s–convex. An. Univ. Oradea Fasc. Mat. 2018, 25, 185–212. [Google Scholar]
- Meftah, B.; Kashuri, A. Some new type integral inequalities for approximately harmonic h–convex functions. Mat. Bilten. 2020, 44, 13–36. [Google Scholar]
- Sarikaya, M.Z. On the Ostrowski-type integral inequality. Acta Math. Univ. Comenian. (N.S.) 2010, 79, 129–134. [Google Scholar]
- Cortez, M.J.V.; García, C.; Kashuri, A.; Hernández, J.E.H. New Ostrowski-type inequalities for coordinated (s,m)–convex functions in the second sense. Appl. Math. Inf. Sci. 2019, 13, 821–829. [Google Scholar]
- Alomari, M.; Darus, M.; Dragomir, S.S. Inequalities of Hermite–Hadamard’s type for functions whose derivatives absolute values are quasi-convex. Tamkang J. Math. 2010, 41, 353–359. [Google Scholar] [CrossRef] [Green Version]
- Alomari, M.; Darus, M. Some Ostrowski type inequalities for quasi-convex functions with applications to special means. RGMIA Res. Rep. Coll. 2010, 13, 9. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; North-Holland Mathematics Studies, 204; Elsevier Science B.V.: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Khalil, R.; Horani, M.A.; Yousef, A.; Sababheh, M. A new definition of fractional derivative. J. Comput. Appl. Math. 2014, 264, 65–70. [Google Scholar] [CrossRef]
- Almeida, R.; Guzowska, M.; Odzijewicz, T. A remark on local fractional calculus and ordinary derivatives. Open Math. 2016, 14, 1122–1124. [Google Scholar] [CrossRef]
- Dragomir, S.S. Ostrowski-type inequalities for generalized Riemann–Liouville fractional integrals of functions with bounded variation. RGMIA Res. Rep. Coll. 2017, 20, Art. 58. [Google Scholar]
- Khan, M.B.; Mohammed, P.O.; Noor, B.; Hamed, Y.S. New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus and related inequalities. Symmetry 2021, 13, 673. [Google Scholar] [CrossRef]
- Kashuri, A.; Rassias, T.M. Some new Ostrowski type integral inequalities via general fractional integrals (Chapter 8). In Computational Mathematics and Variational Analysis; Daras, N., Rassias, T., Eds.; Springer: Berlin/Heidelberg, Germany, 2020; Volume 159, pp. 135–151. [Google Scholar]
- Mohammed, P.O.; Aydi, H.; Kashuri, A.; Hamed, Y.S.; Abualnaja, K.M. Midpoint inequalities in fractional calculus defined using positive weighted symmetry function kernels. Symmetry 2021, 13, 550. [Google Scholar] [CrossRef]
- Meftah, B. Fractional Ostrowski-type inequalities for functions whose first derivatives are ϕ–preinvex. J. Adv. Math. Stud. 2017, 10, 335–347. [Google Scholar]
- Meftah, B. Fractional Ostrowski-type inequalities for functions whose first derivatives are s–preinvex in the second sense. Int. J. Anal. App. 2017, 15, 146–154. [Google Scholar]
- Alqudah, M.A.; Kashuri, A.; Mohammed, P.O.; Raees, M.; Abdeljawad, T.; Anwar, M.; Hamed, Y.S. On modified convex interval valued functions and related inclusions via the interval valued generalized fractional integrals in extended interval space. AIMS Math. 2021, 6, 4638–4663. [Google Scholar] [CrossRef]
- Mohammed, P.O.; Abdeljawad, T.; Alqudah, M.A.; Jarad, F. New discrete inequalities of Hermite-Hadamard type for convex functions. Adv. Differ. Equ. 2021, 2021, 122. [Google Scholar] [CrossRef]
- Meftah, B.; Azaizia, A. Fractional Ostrowski-type inequalities for functions whose first derivatives are MT–preinvex. Rev. Matemáticas Univ. Atlántico Páginas 2019, 6, 33–43. [Google Scholar]
- Mohammed, P.O.; Abdeljawad, T. Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel. Adv. Differ. Equ. 2020, 2020, 363. [Google Scholar] [CrossRef]
- Kashuri, A.; Meftah, B.; Mohammed, P.O. Some weighted Simpson type inequalities for differentiable s-convex functions and their applications. J. Frac. Calc. Nonlinear Sys. 2021, 1, 75–94. [Google Scholar] [CrossRef]
- Mohammed, P.O.; Abdeljawad, T. Opial integral inequalities for generalized fractional operators with nonsingular kernel. J. Inequal. Appl. 2020, 2020, 148. [Google Scholar] [CrossRef]
- Mohammed, P.O.; Abdeljawad, T.; Zeng, S.; Kashuri, A. Fractional Hermite–Hadamard integral inequalities for a new class of convex functions. Symmetry 2020, 12, 1485. [Google Scholar] [CrossRef]
- Set, E. New inequalities of Ostrowski-type for mappings whose derivatives are s–convex in the second sense via fractional integrals. Comput. Math. Appl. 2012, 63, 1147–1154. [Google Scholar] [CrossRef] [Green Version]
- Agarwal, R.P.; Luo, M.J.; Raina, R.K. On Ostrowski-type inequalities. Fasc. Math. 2016, 56, 5–27. [Google Scholar] [CrossRef]
- Baleanu, D.; Kashuri, A.; Mohammed, P.O.; Meftah, B. General Raina fractional integral inequalities on coordinates of convex functions. Adv. Differ. Equ. 2021, 82, 23. [Google Scholar]
- Alb Lupaş, A. Inequalities for Analytic Functions Defined by a Fractional Integral Operator (Chapter 36). In Frontiers in Functional Equations and Analytic Inequalities; Anastassiou, G., Rassias, J., Eds.; Springer: Berlin/Heidelberg, Germany, 2019; pp. 731–745. [Google Scholar]
- Dragomir, S.S.; Cerone, P.; Roumeliotis, J.; Wang, S. A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis. Bull. Math. Soc. Sci. Math. Roum. (N.S.) 1999, 42, 301–314. [Google Scholar]
| Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).