Some new notions of fractional Hermite-Hadamard type inequalities involving applications to the physical sciences

Volume 33, Issue 1, pp 27--41 http://dx.doi.org/10.22436/jmcs.033.01.03

Publication Date: November 16, 2023 Submission Date: July 24, 2023 Revision Date: August 26, 2023 Accteptance Date: October 14, 2023

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 232 Downloads
• 865 Views

Authors

H. Ahmad - Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina, 42210 , Saudi Arabia. - Near East University, Operational Research Center in Healthcare, TRNC Mersin 10, Nicosia, 99138, Turkey. - Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon. R. B. Khokhar - Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan. M. Suleman - Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan. M. Tariq - Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan. S. K. Ntouyas - Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece. J. Tariboon - Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand.

Abstract

The term convexity in the frame of fractional calculus is a well-established concept that has assembled significant attention in mathematics and various scientific disciplines for over a century. It offers valuable insights and results in diverse fields, along with practical applications due to its geometric interpretation. Moreover, convexity provides researchers with powerful tools and numerical methods for addressing extensive interconnected problems. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we construct new fractional identities for differentiable preinvex functions to strengthen the recently assigned approach even more. Then utilizing these identities, some generalizations of the Hermite-Hadamard type inequality involving generalized preinvexities in the frame of fractional integral operator, namely Riemann-Liouville (R-L) fractional integrals are explored. Finally, we examined some applications to the \(q\)-digamma and Bessel functions via the established results. We used fundamental methods to arrive at our conclusions. We anticipate the techniques and approaches addressed by this study will further pique and spark the researcher's interest.

Share and Cite

Keywords

• Convex function
• invex sets
• preinvex functions
• Holder's inequality
• power mean inequality

MSC

•  26A51
•  26A33
•  26D07
•  26D10
•  26D15

References

• [1] P. Agarwal, J. Choi, R. B. Paris, Extended Riemann-Liouville fractional derivative operator and its applications, J. Nonlinear Sci. Appl., 8 (2015), 451–466
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] T. Antczak, Mean value in invexity analysis, Nonlinear Anal., 60 (2005), 1473–1484
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] A. Barani, A. G. Ghazanfari, S. S. Dragomir, Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex, J. Inequal. Appl., 2012 (2012), 9 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] D. Breaz, C. Yildiz, L. I. Cotirla, G. Rahman, B. Yerg¨oz, New Hadamard type inequalities for modified h-convex functions, Fractal Fract., 7 (2023), 12 pages
  • View Article
  • Google Scholar


• [5] H. Budak, M. A. Ali, M. Tarhanaci, Some new quantum Hermite-Hadamard-like inequalities for coordinated convex functions, J. Optim. Theory Appl., 186 (2020), 899–910
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] H. Budak, T. Tunc¸, M. Z. Sarikaya, Fractional Hermite-Hadamard-type inequalities for interval-valued functions, Proc. Amer. Math. Soc., 148 (2020), 705–718
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] S. I. Butt, L. Horv´ath, D. Peˇcari´c, J. Pe˘cari´, Cyclic improvements of Jensen’s inequalities—cyclic inequalities in information theory, ELEMENT, Zagreb (2020)
  • View Article
  • Google Scholar
  • MATH


• [8] S. I. Butt, M. Nadeem, G. Farid, On Caputo fractional derivatives via exponential s-convex functions, Turk. J. Sci., 5 (2020), 140–146
  • View Article
  • Google Scholar


• [9] S. I. Butt, D. Peˇcari´c, J. Peˇcari´, Several Jensen-Gr¨uss inequalities with applications in information theory, Ukrainian Math. J., 74 (2023), 1888–1908
  • View Article
  • Google Scholar
  • MATH


• [10] S. I. Butt, S. Yousaf, A. O. Akdemir, M. A. Dokuyucu, New Hadamard-type integral inequalities via a general form of fractional integral operators, Chaos Solitons Fractals, 148 (2021), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, In: Fractals and fractional calculus in continuum mechanics (Udine, (1996), Springer, Vienna, 378 (1997), 223–276
  • View Article
  • Google Scholar


• [12] ˙I. ˙Is¸can, A new generalization of some integral inequalities for (,m)-convex functions, Math. Sci., 7 (2013), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] I. ˙Is¸can, Some new Hermite-Hadamard type inequalities for geometrically convex functions, Math. Sci., 1 (2013), 86–91
  • View Article


• [14] S. Jain, K. Mehrez, D. Baleanu, P. Agarwal, Certain Hermite–Hadamard inequalities for logarithmically convex functions with applications, Mathematics, 7 (2019), 1–12
  • View Article
  • Google Scholar


• [15] M. Kadakal, ˙I. ˙Is¸can, H. Kadakal, K. Bekar, On improvements of some integral inequalities, Honam Math. J., 43 (2021), 441–452
  • View Article
  • MathSciNet
  • Google Scholar


• [16] M. B. Khan, M. A. Noor, K. I. Noor, Y.-M. Chu, New Hermite-Hadamard-type inequalities for-convex fuzzy-intervalvalued functions, Adv. Difference Equ., 2021 (2021), 20 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [17] S. Khan, M. Adil Khan, S. I. Butt, Y.-M. Chu, A new bound for the Jensen gap pertaining twice differentiable functions with applications, Adv. Difference Equ., 2020 (2020), 11 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] U. S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147 (2004), 137–146
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] J.-B. Liu, S. I. Butt, J. Nasir, A. Aslam, A. Fahad, J. Soontharanon, Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator, AIMS Math., 7 (2022), 2123–2140
  • View Article
  • MathSciNet
  • Google Scholar


• [20] N. Mehmood, S. I. Butt, D. Peˇcari´c, J. Peˇcari´, Generalizations of cyclic refinements of Jensen’s inequality by Lidstone’s polynomial with applications in information theory, J. Math. Inequal., 14 (2020), 249–271
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] S. K. Mishra, G. Giorgi, Invexity and Optimization, Springer-Verlag, Berlin (2008)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] S. R. Mohan, S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901–908
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] M. A. Noor, Hermite-Hadamard integral inequalities for log-preinvex functions, J. Math. Anal. Approx. Theory, 2 (2007), 126–131
  • View Article
  • Google Scholar
  • MATH


• [24] M. A. Noor, Hadamard integral inequalities for product of two preinvex function, Nonlinear Anal. Forum, 14 (2009), 167–173
  • View Article
  • Google Scholar
  • MATH


• [25] M. A. Noor, K. I. Noor, Higher order strongly generalized convex functions, Appl. Math. Inf. Sci., 14 (2020), 133–139
  • View Article
  • MathSciNet
  • Google Scholar


• [26] M. A. Noor, K. I. Noor, M. U. Awan, Geometrically relative convex functions, Appl. Math. Inf. Sci., 8 (2014), 607–616
  • View Article
  • MathSciNet
  • Google Scholar


• [27] M. A. Noor, K. I. Noor, M. U. Awan, Generalized convexity and integral inequalities, Appl. Math. Inf. Sci., 9 (2015), 233–243
  • View Article
  • MathSciNet
  • Google Scholar


• [28] M. E. O¨ zdemir, S. I. Butt, B. Bayraktar, J. Nasir, Several integral inequalities for (, s,m)-convex functions, AIMS Math., 5 (2020), 3906–3921
  • View Article
  • MathSciNet
  • Google Scholar


• [29] R. Pini, Invexity and generalized convexity, Optimization, 22 (1991), 513–525
  • View Article
  • MathSciNet
  • Google Scholar


• [30] T. Rasheed, S. I. Butt, D. Peˇcari´c, J. Peˇcari´c, Generalized cyclic Jensen and information inequalities, Chaos Solitons Fractals, 163 (2022), 9 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] T. Rasheed, S. I. Butt, D. Peˇcari´c, J. Peˇcari´c, A. O. Akdemir, Uniform treatment of Jensen’s inequality by Montgomery identity, J. Math., 2021 (2021), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] S. K. Sahoo, H. Ahmad, M. Tariq, B. Kodamasingh, H. Aydi, M. De la Sen, Hermite-Hadamard type inequalities involving k-fractional operator for (h,m)-convex functions, Symmetry, 13 (2021), 1–18
  • View Article
  • Google Scholar


• [33] E. Set, S. I. Butt, A. O. Akdemir, A. Karaoˇ glan, T. Abdeljawad, New integral inequalities for differentiable convex functions via Atangana-Baleanu fractional integral operators, Chaos Solitons Fractals, 143 (2021), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [34] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge (1995)
  • View Article
  • Google Scholar


• [35] T. Weir, B. Mond, Pre-inven functions in multiple objective optimization, J. Math. Anal. Appl., 136 (1988), 29–38
  • View Article
  • Google Scholar


• [36] S. Wu, M. U. Awan, M. A. Noor, K. I. Noor, S. Iftikhar, On a new class of convex functions and integral inequalities, J. Inequal. Appl., 2019 (2019), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [37] X. M. Yang, D. Li, On properties of preinvex functions, J. Math. Anal. Appl., 256 (2001), 229–241
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH