Abstract
Building upon previous research in conformable fractional calculus, this study introduces a novel identity. Using this identity as a foundation, we derive a set of conformable fractional Milne-type inequalities specifically designed for differentiable convex functions. The obtained results recover some existing inequalities in the literature by fixing some parameters. These novel contributions aim to enrich the analytical tools available for studying convex functions within the realm of conformable fractional calculus. The derived inequalities reflect an inherent symmetry characteristic of the Milne formula, further illustrating the balanced and harmonious mathematical structure within these frameworks. We provide a thorough example with graphical representations to support our findings, offering both numerical insights and visual confirmation of the established inequalities.
Keywords:
conformable fractional integral operator; Milne-type inequalities; convex functions; Hölder inequality; power mean inequality MSC:
65D32; 26A45; 26D10; 26D15
1. Introduction
Convexity is a simple and natural notion. A function is characterized as convex when it satisfies the condition that, for all and , the inequality
holds, as outlined in [].
The concept of convexity stands as a cornerstone with profound implications across various disciplines, each underscored by extensive research and practical applications. In mathematics, convexity provides a fundamental framework for analyzing the geometrical properties of sets and functions, shaping the foundation of optimization theory. This mathematical principle extends its influence into economics, where convex optimization plays a crucial role in modeling and solving economic problems, as discussed in works such as []. In optimization, convexity is pivotal for the development of efficient algorithms that find optimal solutions, with references to key algorithms available in []. Furthermore, convexity is a central concept in game theory, providing insights into strategic interactions among rational decision-makers, as explored in seminal works such as []. The versatility of convexity makes it an indispensable notion, contributing significantly to mathematical theories and finding practical applications in decision sciences, resource allocation, and risk management [].
Integral inequalities, crucial in diverse mathematical and scientific domains, share a pivotal connection with convexity. This relationship has led to the establishment of numerous results related to various quadrature formulas; see [,,] and the references therein.
The fundamental inequality associated with the concept of convexity is the Hermite–Hadamard inequality [], formulated as follows. Let be a convex function on ; then, we have
Regarding historical considerations about inequality (1), we refer readers to [,,] and the references therein.
Among the three-point Newton–Cotes quadrature formulas, Milne’s formula stands out as a specific expression. It is defined by the following approximation of the integral of a function over a given interval:
where denotes the approximation error; see [].
In [], Djenaoui et al. established Milne-type inequalities for differentiable convex functions, as follows:
and
Fractional calculus, an area of mathematical analysis that extends traditional differentiation and integration to non-integer orders, has become increasingly significant across various scientific disciplines. This approach serves as a robust mathematical framework for describing systems with nonlocal or memory-dependent behavior. Its utility ranges from the analysis of anomalous diffusion in physics [] to modeling intricate biological processes []. As researchers delve deeper into its intricacies, fractional calculus continues to prove invaluable in understanding and characterizing phenomena that classical methods may overlook.
Definition 1
([]). Let . The Riemann–Liouville fractional integrals and of order with are defined by
respectively, where is the gamma function and .
In the realm of fractional calculus, several scholars have devoted their endeavors to establishing a variety of integral inequalities. Noteworthy contributions in this regard can be found in [,,,,,], along with the additional references contained therein.
In [], Budak et al. extended the result obtained in [] to Riemann–Liouville integral operators, as follows:
where is convex on .
Other results can be discussed through Hölder’s inequality and the power mean inequality.
Theorem 1
(Hölder inequality []). Let and . If f and g are real functions defined on and if and are integrable functions on , then
Theorem 2
(Power Mean Integral inequality []). Let . If f and g are real functions defined on and if and are integrable functions on , then
Lemma 1
(Discrete Power Mean inequality []). For any and , we have
To better describe certain phenomena that classical fractional operators fail to model, several new conformable fractional operators have been introduced [,]. Among these, the operator proposed by Jarad et al. in [] stands out as a generalization of various operators, including the Riemann–Liouville and Hadamard operators.
Definition 2
([]). Let . The left- and right-sided conformable fractional integral operators of order with and are expressed as follows:
In the wake of the introduction of these novel operators, a plethora of research endeavors has been undertaken to establish inequalities tailored to this class of integrals. Researchers have explored diverse avenues, contributing to the development of a rich landscape of inequalities. Set et al. laid the foundation by presenting Hermite–Hadamard and trapezium inequalities applicable to differentiable convex functions in []. Subsequently, Ostrowski-type inequalities were derived in [], expanding the toolkit for analyzing this class of fractional integrals. Hyder et al. made significant contributions by introducing midpoint-type inequalities in []. In [], Kara et al. extended these efforts by providing midpoint-type and trapezoid-type inequalities specifically tailored for twice-differentiable convex functions.
Further variations and insights into this realm were presented by Hezenci et al. in [], where Simpson-type inequalities were established, while Ünal et al. contributed Simpson second formula inequalities in []. Rashid et al. explored the Minkowski inequality in [], adding another dimension to the spectrum of inequalities for conformable fractional integrals. Rahman et al. delved into Grüss inequality in [] and Chebyshev inequality in [], further enriching the landscape of mathematical tools applicable to this type of integrals. Nisar et al. continued the exploration of the Minkowski inequality in [], providing additional insights into its applicability.
This research focus was extended to explore specific inequalities, such as the Hermite–Jensen–Mercer inequality investigated by Butt et al. in [] and the examination of Pachpatte inequality by Akdemir et al. in []. These efforts have collectively contributed to a comprehensive understanding of inequalities related to conformable fractional integrals. For those interested in delving further into this topic, additional related works are available for reference in [,,,,,,,].
Building upon the insights gleaned from the previously mentioned works, particularly those outlined in [,], our current study introduces a novel identity. This identity, rooted in the symmetric principles inherent in the Milne formula, serves as the foundation for establishing a set of new conformable fractional Milne-type inequalities designed for differentiable convex functions. The symmetry in this context not only enhances the mathematical elegance of these inequalities, but also reinforces their theoretical validity. To validate the precision of our findings, we offer a comprehensive example complemented by graphical representations that substantiate the obtained outcomes and visually demonstrate the symmetrical nature of these results.
The rest of this paper is structured as follows. In Section 2, we introduce a new identity, from which we proceed to derive conformable fractional integral inequalities. An illustrative example is presented to demonstrate the accuracy of the obtained results. Section 3 discusses applications to composite quadrature formulas. Finally, the conclusion is presented in Section 4.
2. Main Results
Lemma 2.
Let be a differentiable function on I, with . If , then the following equality holds for and :
where
Proof.
Let
and
Integrating by parts , we obtin
Similarly, we have
Theorem 3.
Let be as in Lemma 2. If is convex on , then for and we have
where is defined by (3) and is the Beta function.
Proof.
Taking the absolute value on both sides of (2) and then using the convexity of , we obtain
where we have used
and
The proof is finished. □
Remark 1.
Corollary 1.
In Theorem 3, using the convexity of , i.e., , we obtain
Remark 2.
By taking , Inequality (9) is reduced to the second inequality of Corollary 2.4 from []. The same result was obtained by Budak et al. in Remark 1 from [].
Corollary 2.
Taking in Theorem 3, we obtain
where
Corollary 3.
Now, we present an illustrative example supported by graphical representations to substantiate our findings. It is crucial to mention that the figures were created using Matlab, with red representing the right-hand side of the respective inequalities and blue representing the left-hand side.
Example 1.
Consider the function defined on the interval by . This function meets the assumptions of our study, as its derivative is convex over the interval .
From Theorem 3, we obtain the following inequality, illustrated in Figure 1:

Figure 1.
Illustration for Theorem 3.
These representations show a consistent trend where the right-hand side is greater than the left-hand side, affirming the accuracy of our results.
Theorem 4.
Let be as in Lemma 2. If is convex on , then for all and we have
where is defined as (3), with and is the Beta function.
Proof.
Taking the absolute value on both sides of (2) and then using Hölder’s inequality and the fact that is convex, we deduce
The proof is finished. □
Remark 3.
Corollary 4.
In Theorem 4, using the convexity of , we obtain
Remark 4.
Corollary 5.
In Corollary 4, using the discrete power mean inequality, we obtain
Corollary 6.
In Corollary 5, taking , we obtain
Corollary 7.
Corollary 8.
In Corollary 7, using the convexity of , we obtain
where is defined by (11) and is the hypergeometric function.
Corollary 9.
In Corollary 8, using the discrete power mean inequality, we obtain
where is defined by (11) and is the hypergeometric function.
Theorem 5.
Let be as in Lemma 2. If is convex on , then for all and we have
where is defined by (3), and is the Beta function.
Proof.
Taking the absolute value on both sides of (2) and then using power mean inequality and convexity of , we deduce
where we have used (7), (8), and
The proof is finished. □
Remark 5.
Corollary 10.
In Theorem 5, using the convexity of , we obtain
Remark 6.
Corollary 11.
In Corollary 10, using the discrete power mean inequality, we obtain
Corollary 12.
Corollary 13.
Corollary 14.
3. Applications
Application to Composite Quadrature Formula
Let be the partition of the interval such that , taking the quadrature formula into consideration; then,
where
with denoting the associated approximation error.
Proposition 1.
Let be as in Theorem 3; then, we have
Proof.
By applying inequality Equation (12) with to the partition of subintervals (), we obtain
We reach the necessary result by multiplying both sides of the aforementioned inequality by , summing the generated inequalities for all , and applying the triangular inequality. □
Proposition 2.
Let be as in Theorem 4; then, we have
Proof.
By applying inequality (19) with to the partition of subintervals (), we obtain
We reach the necessary result by multiplying both sides of the aforementioned inequality by , summing the generated inequalities for all , and applying the triangular inequality. □
Proposition 3.
Let be as in Theorem 5; then, we have
Proof.
By applying inequality (25) with to the partition of subintervals (), we obtain
We reach the necessary result by multiplying both sides of the aforementioned inequality by , summing the generated inequalities for all , and applying the triangular inequality. □
4. Conclusions
In summary, our efforts have made a significant contribution to the field of conformable fractional calculus. The introduction of a novel symmetrical identity followed by the derivation of multiple Milne-type inequalities tailored for differentiable convex functions highlights the innovative nature of our work. The illustrative example serves to visually elucidate the underlying concepts of the derived inequalities through graphical representations. This research adds valuable insights to the advancing field of conformable fractional calculus, demonstrating its potential applications in mathematical analysis and modeling. The presented results not only contribute to the current understanding, but can pave the way for future investigations in this promising area of study.
Author Contributions
Conceptualization, R.Y., H.X. and A.L.; methodology, R.Y. and W.S.; software, A.L.; validation, H.X. and B.M.; formal analysis, R.Y.; investigation, R.Y. and W.S.; resources, A.L.; data curation, R.Y. and H.X.; writing—original draft preparation, R.Y. and A.L.; writing—review and editing, R.Y., H.X. and W.S.; visualization, A.L. and B.M.; supervision, B.M.; project administration, H.X. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Data sharing is not relevant to this article, as there was no generation or analysis of new data during the course of this study.
Conflicts of Interest
The authors declare that they have no conflict of interest related to this research.
References
- Jensen, J.L.W.V. Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Math. 1906, 30, 175–193. [Google Scholar] [CrossRef]
- Rockafellar, R.T.; Wets, R.J.B. Variational Analysis; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2009; Volume 317. [Google Scholar]
- Boyd, S.; Vandenberghe, L. Convex Optimization; Cambridge University Press: Cambridge, UK, 2004; pp. 14–716. [Google Scholar]
- Osborne, M.J.; Rubinstein, A. A Course in Game Theory; MIT Press: Cambridge, MA, USA, 1994. [Google Scholar]
- Khan, A.M.; Khurshid, Y.; Du, T.S.; Chu, Y.M. Generalization of Hermite-Hadamard type inequalities via conformable fractional integrals. J. Funct. Spaces 2018, 2018, 5357463. [Google Scholar]
- Kashuri, A.; Du, T.S.; Liko, R. On some new integral inequalities concerning twice differentiable generalized relative semi–(m, h)–preinvex mappings. Stud. Univ. Babes-Bolyai Math. 2019, 64, 43–61. [Google Scholar] [CrossRef]
- Meftah, B.; Lakhdari, A.; Saleh, W.; Kiliçman, A. Some new fractal Milne type integral inequalities via generalized convexity with applications. Fractal Fract. 2023, 7, 166. [Google Scholar] [CrossRef]
- Niculescu, C.P.; Persson, L.-E. Convex functions and their applications. A contemporary approach. In CMS Books in Mathematics/Ouvrages de Mathé Matiques de la SMC; Springer: New York, NY, USA, 2006; Volume 23. [Google Scholar]
- Dragomir, S.S.; Pearce, C.E.M. Selected Topics on Hermite-Hadamard Inequalities and Applications; RGMIA Monographs, Victoria University: Melbourne, VI, USA, 2000. [Google Scholar]
- Mitrinović, D.S.; Lacković, I.B. Hermite and convexity. Aequationes Math. 1985, 28, 229–232. [Google Scholar] [CrossRef]
- Booth, A.D. Numerical Methods, 3rd ed.; Butterworths: La Canada Flintrige, CA, USA, 1966. [Google Scholar]
- Djenaoui, M.; Meftah, B. Milne type inequalities for differentiable s-convex functions. Honam Math. J. 2022, 44, 325–338. [Google Scholar]
- Ullah, M.I.; Ain, Q.T.; Khan, A.; Abdeljawad, T.; Alqudah, M.A. Fractional approach to solar heating model using extended ODE system. Alex. Eng. J. 2023, 81, 405–418. [Google Scholar] [CrossRef]
- Thirthar, A.A.; Nazmul, S.K.; Mondal, B.; Alqudah, M.A.; Abdeljawad, T. Utilizing memory effects to enhance resilience in disease-driven prey-predator systems under the influence of global warming. J. Appl. Math. Comput. 2023, 69, 4617–4643. [Google Scholar] [CrossRef]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional integrals and derivatives. In Theory and Applications; Edited and with a foreword by S. M. Nikol’skiĭ; Translated from the 1987 Russian Original; Revised by the authors; Gordon and Breach Science Publishers: Yverdon, Switzerland, 1993. [Google Scholar]
- Kashuri, A.; Meftah, B.; Mohammed, P.O.; Lupa, A.A.; Abdalla, B.; Hamed, Y.S.; Abdeljawad, T. Fractional weighted Ostrowski-Type inequalities and their applications. Symmetry 2021, 13, 968. [Google Scholar] [CrossRef]
- Lakhdari, A.; Meftah, B. Some fractional weighted trapezoid type inequalities for preinvex functions. Int. J. Nonlinear Anal. Appl. 2022, 13, 3567–3587. [Google Scholar]
- Meftah, B. Fractional Hermite-Hadamard type integral inequalities for functions whose modulus of derivatives are co-ordinated log-preinvex. Punjab Univ. J. Math. 2019, 51, 21–37. [Google Scholar]
- Saleh, W.; Lakhdari, A.; Kiliçman, A.; Frioui, A.; Meftah, B. Some new fractional Hermite-Hadamard type inequalities for functions with co-ordinated extended (s,m)-prequasiinvex mixed partial derivatives. Alex. Eng. J. 2023, 72, 261–267. [Google Scholar] [CrossRef]
- Saleh, W.; Lakhdari, A.; Abdeljawad, T.; Meftah, B. On fractional biparameterized Newton-type inequalities. J. Inequalities Appl. 2023, 2023, 122. [Google Scholar] [CrossRef]
- Budak, H.; Kösem, P.; Kara, H.H. On new Milne-type inequalities for fractional integrals. J. Inequal. Appl. 2023, 2023, 10. [Google Scholar] [CrossRef]
- Mitrinović, D.S.; Pečarić, J.E.; Fink, A.M. Classical and new inequalities in analysis. In Mathematics and its Applications (East European Series); Kluwer Academic Publishers Group: Dordrecht, Germany, 1993; Volume 61. [Google Scholar]
- Bullen, P.S.; Mitrinović, S.D.; Vasić, P.M. Means and their inequalities. In Mathematics and its Applications (East European Series); Translated and revised from the Serbo-Croatian; D. Reidel Publishing Co.: Dordrecht, The Netherlands, 1988. [Google Scholar]
- Hyder, A.; Soliman, A.H. A new generalized θ-conformable calculus and its applications in mathematical physics. Phys. Scr. 2020, 96, 15208. [Google Scholar] [CrossRef]
- Zhao, D.; Luo, M. General conformable fractional derivative and its physical interpretation. Calcolo 2017, 54, 903–917. [Google Scholar] [CrossRef]
- Jarad, F.; Uğurlu, E.; Abdeljawad, T.; Baleanu, D. On a new class of fractional operators. Adv. Differ. Equ. 2017, 2017, 247. [Google Scholar] [CrossRef]
- Set, E.; Choi, J.; Gözpınar, A. Hermite-Hadamard type inequalities involving nonlocal conformable fractional integrals. Malays. J. Math. Sci. 2021, 15, 33–43. [Google Scholar]
- Set, E.; Akdemir, A.O.; Gözpınar, A.; Jarad, F. Ostrowski type inequalities via new fractional conformable integrals. AIMS Math. 2019, 4, 1684–1697. [Google Scholar] [CrossRef]
- Hyder, A.A.; Budak, H.; Almoneef, A.A. Further midpoint inequalities via generalized fractional operators in Riemann–Liouville sense. Fractal Fract. 2022, 6, 496. [Google Scholar] [CrossRef]
- Kara, H.; Budak, H.; Etemad, S.; Rezapour, S.; Ahmad, H.; Kaabar, M.K.A. A study on the new class of inequalities of midpoint-type and trapezoidal-type based on twice differentiable functions with conformable operators. J. Funct. Spaces 2023, 2023, 4624604. [Google Scholar] [CrossRef]
- Hezenci, F.; Budak, H. Simpson-type inequalities for conformable fractional operators with respect to twice-differentiable functions. J. Math. Ext. 2023, 17. [Google Scholar]
- Ünal, C.; Hezenci, F.; Budak, H. Conformable fractional Newton-type inequalities with respect to differentiable convex functions. J. Inequal. Appl. 2023, 2023, 85. [Google Scholar] [CrossRef]
- Rashid, S.; Akdemir, A.O.; Nisar, K.S.; Abdeljawad, T.; Rahman, G. New generalized reverse Minkowski and related integral inequalities involving generalized fractional conformable integrals. J. Inequal. Appl. 2020, 2020, 177. [Google Scholar] [CrossRef]
- Rahman, G.; Nisar, K.S.; Qi, F. Some new inequalities of the Grüss type for conformable fractional integrals. AIMS Math. 2018, 3, 575–583. [Google Scholar] [CrossRef]
- Rahman, G.; Ullah, Z.; Khan, A.; Set, E.; Nisar, K.S. Certain Chebyshev-type inequalities involving fractional conformable integral operators. Mathematics 2019, 7, 364. [Google Scholar] [CrossRef]
- Nisar, K.S.; Tassaddiq, A.; Gauhar, R.; Khan, A. Some inequalities via fractional conformable integral operators. J. Inequal. Appl. 2019, 2019, 217. [Google Scholar] [CrossRef]
- Butt, S.I.; Akdemir, A.O.; Nasir, J.; Jarad, F. Some Hermite-Jensen-Mercer like inequalities for convex functions through a certain generalized fractional integrals and related results. Miskolc Math. Notes 2020, 21, 689–715. [Google Scholar] [CrossRef]
- Akdemir, A.O.; Ekinci, A.; Set, E. Conformable fractional integrals and related new integral inequalities. J. Nonlinear Convex Anal. 2017, 18, 661–674. [Google Scholar]
- AlNemer, G.; Kenawy, M.; Zakarya, M.; Cesarano, C.; Rezk, H.M. Generalizations of Hardy’s type inequalities via conformable calculus. Symmetry 2021, 13, 242. [Google Scholar] [CrossRef]
- AlNemer, G.; Kenawy, M.R.; Rezk, H.M.; El-Deeb, A.A.; Zakarya, M. Fractional Leindler’s Inequalities via Conformable Calculus. Symmetry 2022, 14, 1958. [Google Scholar] [CrossRef]
- AlNemer, G.; Zakarya, M.; Butush, R.; Rezk, H.M. Some New Bennett–Leindler Type Inequalities via Conformable Fractional Nabla Calculus. Symmetry 2022, 14, 2183. [Google Scholar] [CrossRef]
- Hezenci, F.; Budak, H. Novel results on trapezoid-type inequalities for conformable fractional integrals. Turkish J. Math. 2023, 47, 425–438. [Google Scholar] [CrossRef]
- Hezenci, F.; Kara, H.; Budak, H. Conformable fractional versions of Hermite-Hadamard-type inequalities for twice-differentiable functions. Bound. Value Probl. 2023, 2023, 48. [Google Scholar] [CrossRef]
- Hyder, A.A.; Barakat, M.A.; Fathallah, A.; Cesarano, C. Further Integral Inequalities through Some Generalized Fractional Integral Operators. Fractal Fract. 2021, 5, 282. [Google Scholar] [CrossRef]
- Rezk, H.M.; Albalawi, W.; El-Hamid, H.A.A.; Saied, A.I.; Bazighifan, O.; Mohamed, M.S.; Zakarya, M. Hardy-Leindler-type inequalities via conformable delta fractional calculus. J. Funct. Spaces 2022, 2022, 2399182. [Google Scholar] [CrossRef]
- Zakarya, M.; Altanji, M.; AlNemer, G.; El-Hamid, H.A.A.; Cesarano, C.; Rezk, H.M. Fractional reverse coposnís inequalities via conformable calculus on time scales. Symmetry 2021, 13, 542. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 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/).