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Caputo fractional derivative Hadamard inequalities for strongly \(m\)-convex functions. (English) Zbl 1475.26005

The authors derive and prove two versions of Hadamard-type inequalities using the concept of Caputo fractional derivatives and strongly \(m\)-convex functions. Several consequences of their results are pointed out. Furthermore, error bounds of fractional Hadamard inequalities are also derived, proved and discussed.

MSC:

26A33 Fractional derivatives and integrals
26D10 Inequalities involving derivatives and differential and integral operators
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[1] Polyak, B. T., Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Mathematics Doklady, 7, 72-75 (1966) · Zbl 0171.09501
[2] Makò, J.; Hézy, A., On strongly convex functions, Carpathian Journal of Mathematics, 32, 1, 87-95 (2016) · Zbl 1389.39034
[3] Merentes, N.; Nikodem, K., Remarks on strongly convex functions, Aequationes Math, 80, 1-2, 193-199 (2010) · Zbl 1214.26007 · doi:10.1007/s00010-010-0043-0
[4] Nikodem, K.; Rassias, T., On strongly convex functions and related classes of functions, Handbook of Functional Equations. Springer Optimization and Its Applications, vol 95 (2014), New York, NY, USA: Springer, New York, NY, USA · Zbl 1316.26014 · doi:10.1007/978-1-4939-1246-9_16
[5] Nikodem, K.; Páles, Z., Characterizations of inner product spaces by strongly convex functions, Banach Journal of Mathematical Analysis, 5, 1, 83-87 (2011) · Zbl 1215.46016 · doi:10.15352/bjma/1313362982
[6] Vial, J. P., Strong convexity of sets and functions, Journal of Mathematical Economics, 9, 1-2, 187-205 (1982) · Zbl 0479.52005 · doi:10.1016/0304-4068(82)90026-X
[7] Toader, G. H., Some generalisations of the convexity, Proc. Colloq. Approx. Optim
[8] Lara, T.; Merentes, N.; Quintero, R.; Rosales, E., On strongly \(\text{m} \)-convex functions, Mathematica Aeterna, 5, 3, 521-535 (2015)
[9] Agarwal, P.; Jleli, M.; Tomar, M., Certain Hermite-Hadamard type inequalities via generalized \(\text{k} \)-fractional integrals, Journal of Inequalities and Applications, 2017, 1 (2017) · Zbl 1359.26005 · doi:10.1186/s13660-017-1318-y
[10] Chen, F., On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals via two kinds of convexity, Chinese Journal of Mathematics, 2014 (2014) · Zbl 1298.26020 · doi:10.1155/2014/173293
[11] Dong, Y.; Zeb, M.; Farid, G.; Bibi, S., Hadamard inequalities for strongly α, m-convex functions via Caputo fractional derivatives, Journal of Mathematics, 2021 (2021) · Zbl 1477.26032 · doi:10.1155/2021/6691151
[12] Farid, G.; Javed, A.; Rehman, A. U., On Hadamard inequalities for \(n-\) times differentiable functions which are relative convex via Caputo \(\text{k} \)-fractional derivatives, Nonlinear Analysis Forum, 22, 2, 17-28 (2017) · Zbl 1414.26045
[13] Farid, G.; Javed, A.; Rehman, A. U.; Qureshi, M. I., On Hadamard-type inequalities for differentiable functions via Caputok-fractional derivatives, Cogent Mathematics, 4, 1, article 1355429 (2017) · Zbl 1438.26060 · doi:10.1080/23311835.2017.1355429
[14] Farid, G.; Rehman, A. U.; Zahra, M., On Hadamard inequalities for \(\text{k} \)-fractional integrals, Nonlinear Functional Analysis and Applications, 21, 3, 463-478 (2016) · Zbl 1358.26014
[15] Özdemir, M. E.; Akdemri, A. O.; Set, E., On \(\left( h - m\right)-\) convexity and Hadamard-type inequalities, Transylvanian Journal of Mathematics and Mechanics, 8, 1, 51-58 (2016)
[16] Sarikaya, M. Z.; Set, E.; Yaldiz, H.; Başak, N., Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57, 9-10, 2403-2407 (2013) · Zbl 1286.26018 · doi:10.1016/j.mcm.2011.12.048
[17] Waheed, A.; Rehman, A. U.; Qureshi, M. I.; Shah, F. A.; Khan, K. A.; Farid, G., On Caputo k-fractional derivatives and associated inequalities, IEEE Access, 7, 32137-32145 (2019) · doi:10.1109/ACCESS.2019.2902317
[18] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Amsterdam: Elsevier, Amsterdam · Zbl 1092.45003
[19] Dragomir, S. S.; Pearce, C. E. M., Selected topics on Hermite-Hadamard inequalities and applications, Mathematics Preprint Archive, 1, 463-817 (2003)
[20] Farid, G.; Javed, A.; Naqvi, S., Hadamard and Fejer Hadamard inequalities and related results via Caputo fractional derivatives, Bulletin of Mathematical Analysis and Applications, 9, 3, 16-30 (2017) · Zbl 1408.26021
[21] Kang, S. M.; Farid, G.; Nazeer, W.; Naqvi, S., A version of the Hadamard inequality for Caputo fractional derivatives and related results, Journal of Computational Analysis and Applications, 27, 6, 962-972 (2019)
[22] Farid, G.; Javed, A.; Rehman, A. U., Fractional integral inequalities of Hadamard type for \(\text{m} \)-convex functions via Caputo \(\text{k} \)-fractional derivatives, Journal of Fractional Calculus and Applications, 10, 1, 120-134 (2019) · Zbl 1485.26033
[23] Farid, G.; Rehman, A. U.; Bibi, S.; Chu, Y.-M., Refinements of two fractional versions of Hadamard inequalities for Caputo fractional derivatives and related results, Journal of Mathematical Sciences, 5, 1, 1-10 (2021) · doi:10.30538/oms2021.0139
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