Inequalities for strongly convex functions via Atangana-Baleanu Integral Operators

Abstract

In this study, new results are generated for strongly convex functions with the help of Atangana-Baleanu integral operator.

Keywords

• Strongly convex functions
• Ostrowski inequality
• Hölder inequality
• Atangana-Baleanu integral operators
• Normalization function
• Euler Gamma function

Kaynakça

1. Referans1: T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl. 10 (2017), 1098-1107.
2. Referans2 : T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Reports on Mathematical Physics 80 (2017), no. 1, 11-27.
3. Referans3 : A.O. Akdemir, A. Ekinci and E. Set, Conformable fractional integrals and related new integral inequalities, J. Nonlinear Convex Anal. 18 (2017), no. 4, 661--674.
4. Referans4 : A.O. Akdemir, S.I. Butt, M. Nadeem and M.A. Ragusa, New general variants of Chebyshev type inequalities via generalized fractional integral operators, Mathematics 9 (2021), no.2, 122.
5. Referans5 : F.A. Aliev, N.A. Aliev and N.A. Safarova, Transformation of the Mittag-Leffler function to an exponential function and some of its applications to problems with a fractional derivative, Applied and Computational Mathematics 18 (2019), no. 3, 316-325.
6. Referans6 : A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications 505 (2018), 688-706.
7. Referans7 : A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and Application to Heat Transfer Model, Thermal Science 20 (2016), no. 2, 763-769.
8. Referans8 : A. Atangana and J.F.Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos, Solitons & Fractals 114 (2018), 516-535.

9. Referans9 : A.Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons & Fractals 89 (2016), 447-454.
10. Referans10 : M. Avcı Ardıç, A.O. Akdemir and H. Kavurmacı Önalan, Integral inequalities for differentiable s-convex functions in the second sense via Atangana-Baleanu integral operators, Submitted on March 2021.
11. Referans11 : A. Azócar, K. Nikodem and G. Roa, Fejér-type inequalities for strongly convex functions, Annales Mathematicae Silesianae 26 (2012), 43-54.
12. Referans12: M. Klaričić Bakula and K. Nikodem, On the converse Jensen inequality for strongly convex functions, J. Math. Anal. Appl. 434 (2016), 516-522.
13. Referans13 : S.I. Butt, M. Nadeem, S. Qaisar, A.O. Akdemir and T. Abdeljawad, Hermite-Jensen-Mercer type inequalities for conformable integrals and related results, Advances in Difference Equations 2020 (2020), Article Number: 501.
14. Referans14 : S.I. Butt and J. Pecarić, Generalized Hermite-Hadamard's inequality, Proc. A. Razmadze Math. Inst. 163 (2013), 9-27.
15. Referans15 : M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), no. 2, 73-85.
16. Referans16 : P. Cerone, S. S. Dragomir, and J. Roumeliotis, An inequality of Ostrowski-Grüss type for twice differentiable mappings and applications in numerical integration, KYUNGPOOK Math. J. 39 (1999), no. 2, 333-341.
17. Referans17 : X.-L. Cheng, Improvement of some Ostrowski-Grüss type inequalities, Computers & Mathematics with Applications 42 (2001), no. 1-2, 109-114.
18. Referans18 : M. A. Dokuyucu, Caputo and Atangana-Baleanu-Caputo fractional derivative applied to garden equation, Turkish Journal of Science 5 (2020), no. 1, 1-7.
19. Referans19 : M. A. Dokuyucu, Analysis of the Nutrient-Phytoplankton-Zooplankton system with non-local and non-singular kernel, Turkish Journal of Inequalities 4 (2020), no. 1, 58-69.
20. Referans20 : M.A. Dokuyucu, D. Baleanu and E. Celik, Analysis of Keller-Segel model with Atangana-Baleanu fractional derivative, Filomat 32 (2018), no. 16, 5633-5643.
21. Referans21 : S.S. Dragomir and S. Wang, An inequality of Ostrowski-Grüss' type and its applications to the estimation of error bounds for some special means and for some numeraical quadrature rules, Computers & Mathematics with Applications 33 (1997), no.11, 15-20.
22. Referans22 : H. Kavurmaci, M. Avci and M. E. Özdemir, New inequalities of Hermite-Hadamard type for convex functions with applications, Journal of Inequalities and Applications 2011, 2011:86.
23. Referans23 : U. S. Kirmaci, M.Klaričić Bakula, M. E. Özdemir and J. Pečarić, Hadamard-type inequalities for s-convex functions, Applied Mathematics and Computation 193 (2007), no. 1, 26-35.
24. Referans24 : Z. Liu, Some Ostrowski-Grüss type inequalities and applications, Computers & Mathematics with Applications, 53 (2007), no. 1, 73-79.
25. Referans25: Q. Li, M.S. Saleem, P. Yan, M.S. Zahoor and M. Imran, On strongly convex functions via Caputo-Fabrizio-type fractional integral and some applications, Journal of Mathematics 2021 (2021), Article ID 6625597.
26. Referans26: M. Matić, J. Pečarić and N. Ujević, Improvement and further generalization of inequalities of Ostrowski-Grüss type, Computers & Mathematics with Applications 39 (2000), no. 3-4, 161-175.
27. Referans27: N. Merentes and K. Nikodem, Remarks on strongly convex functions, Aequat. Math. 80 (2010), 193-199.
28. Referans28: G.V. Milovanović and J. E. Pečarić, On generalization of the inequality of A. Ostrowski and some related applications, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz. (544-576), 155-158, 1976.
29. Referans29: M. Niezgoda, A new inequality of Ostrowski--Grüss type and applications to some numerical quadrature rules, Computers & Mathematics with Applications, 58 (2009), no. 3, 589-596.
30. Referans30: A. Ostrowski, Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv. 10 (1938), 226-227.
31. Referans31: K.M. Owolabi, Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative, Eur. Phys. J. Plus 133 (2018), Article Number:15.
32. Referans32: M. E. Özdemir, A.O. Akdemir and A. Ekinci, New integral inequalities for co-ordinated convex functions, Fundamentals of Contemporary Mathematical Sciences 2 (2021), no. 1, 52-69.
33. Referans33: M. E. Özdemir, A. Ekinci, A.O. Akdemir, Some new integral inequalities for functions whose derivatives of absolute values are convex and concave, TWMS Journal of Pure and Applied Mathematics, 10 (2019), no. 2, 212-224.
34. Referans34: M. E. Ozdemir, M. A. Latif and A.O.Akdemir, On some Hadamard-type inequalities for product of two h-convex functions on the co-ordinates, Turkish Journal of Science 1 (2016), no. 1, 41-58.
35. Referans35: C.E.M. Pearce, J. Pečarić, N. Ujević and S. Varošanec, Generalizations of some inequalities of Ostrowski Grüss type, Math. Inequal. Appl. 3 (2000), no. 1, 25-34.
36. Referans36: J. Pečarić, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, 1992.
37. Referans37: B. T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl. 7 (1966), 72-75.
38. Referans38: E. Set, New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals, Computers and Mathematics with Applications, 63 (2012), no. 7, 1147-1154.
39. Referans39: E. Set, A.O. Akdemir and F. Özata, Grüss type inequalities for fractional integral operator involving the extended generalized Mittag-Leffler function, Applied and Computational Mathematics, 19 (2020), no. 3, 402-414.
40. Referans40: E. Set, A.O. Akdemir, M.E. Özdemir, Simpson type integral inequalities for convex functions via Riemann-Liouville integrals, Filomat 31 (2017), no. 14, 4415-4420.
41. Referans41: E. Set, M.E. Özdemir, M.Z. Sarıkaya and A.O. Akdemir, Ostrowski's type inequalities for strongly-convex functions, Georgian Mathematical Journal 25 (2012), no. 1.
42. Referans42: N. Ujević, New bounds for the first inequality of Ostrowski-Grüss type and applications, Computers & Mathematics with Applications 46 (2003), no. 2-3, 421-427.
43. Referans43: S. Yang, A unified approach to some inequalities of Ostrowski--Grüss type, Computers & Mathematics with Applications 51 (2006), no. 6-7, 1047-1056 .