Hermite-Hadamard Type Inequalities for Generalized Fractional Integrals via Strongly Convex Functions

Yıl 2019, Cilt: 7 Sayı: 2, 268 - 273, 15.10.2019

Hasan Fehmi Gidergelmez , Abdullah Akkurt , Hüseyin Yıldırım

Öz

In this paper, the authors have obtained some new developments of Hermite-Hadamard type inequalities for generalized fractional integrals defined by Mubeen et. al. \cite{SSM}. In the last part of the article, some results are given with the help of the definition of many fractional integral arising from the generalization.

Anahtar Kelimeler

Strongly convex functions, Fractional integrals, Hermite-Hadamard inequality

Kaynakça

• [1] A. Akkurt, M. E. Yıldırım and H. Yıldırım, On some Integral Inequalities for (k;h)􀀀Riemann-Liouville fractional integral, New Trends in Mathematical Sciences (NTMSCI), 4(1), (2016), 138-146.
• [2] B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. Dokl. 7, 72–75 (1966).
• [3] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Diferential Equations, Elsevier B.V., Amsterdam, Netherlands, 2006.
• [4] N. Merentes and K. Nikodem, Remarks on strongly convex functions. Aequ. Math. 80, 193–199 (2010).
• [5] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach, Linghorne, 1993.
• [6] S. Mubeen, S. Iqbal and M. Tomar, Hermite-Hadamard Type Inequalities via Fractional Integrals of a Function With Respect to Another Function and k􀀀parameter, J. Inequal. Math. Appl. 1 (2017), 1-9.
• [7] M. Zeki Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite-Hadamard’s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, doi:10.1016/j.mcm.2011.12.048, 57 (2013) 2403–2407.
• [8] F. Ertugral, M. Zeki Sarıkaya and H. Budak, On Refinements of Hermite-Hadamard-Fejer Type Inequalities for Fractional Integral Operators, Applications and Applied Mathematics,Vol. 13, Issue 1 (June 2018), pp. 426-442.
• [9] M. Zeki Sarikaya and H. Yildirim, “On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals”, Miskolc Mathematical Notes, 17(2), 1049-1059, 2016.
• [10] H. Budak and M. Zeki Sarikaya, Hermite-Hadamard type inequalities for s-convex mappings via fractional integrals of a function with respect to another function, Fasciculi Mathematici, No.27, pp.25-36, 2016. DOI:10.1515/fascmath-2016-0014.
• [11] M. Zeki Sarikaya and H. Budak, ”Generalized Hermite-Hadamard type integral inequalities for fractional integrals”, FILOMAT, 30:5 (2016), 1315–1326.
• [12] J. Vanterler da C. Sousa, E. Capelas de Oliveira, The Minkowski’s inequality by means of a generalized fractional integral. AIMS Mathematics, 2018, 3(1): 131-147. doi: 10.3934/Math.2018.1.131
• [13] J. Sousa, D. S. Oliveira and E. C. de Oliveira, Gruss-type inequality by mean of a fractional integral, arXiv:1705.00965, (2017).
• [14] E. Set, Z. Dahmani and ˙I. Mumcu, New extensions of Chebyshev type inequalities using generalized Katugampola integrals via Polya-Szego inequality. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(2), 137-144. doi:http://dx.doi.org/10.11121/ijocta.01.2018.00541.
• [15] S.K. Ntouyas, P. Agarwal and J. Tariboon, On P´olya-Szeg¨o and Chebyshev type inequalities involving the Riemann-Liouville fractional integral op-erators.J. Math. Inequal,10(2), 491-504.
• [16] P. Agarwal, M. Jleli and M. Tomar, Certain Hermite-Hadamard type inequalities via generalized k􀀀fractional integrals. J. Inequal. Appl. 2017(1) (2017).
• [17] P. Agarwal J. Tariboon, and S.K. Ntouyas, Some generalized Riemann-Liouville k􀀀fractional integral inequalities. Journal of Inequalities and Applications, 2016, 122. https://doi.org/10.1186/s13660-016-1067-3.
• [18] M. Tomar, S. Mubeen and J. Choi, Certain inequalities associated with Hadamard k-fractional integral operators, Journal of Inequalities and Applications20162016:234, https://doi.org/10.1186/s13660-016-1178-x.
• [19] J. Vanterler da C. Sousa, E. Capelas de Oliveira., A Gronwall inequality and the Cauchy-type problem by means of y􀀀Hilfer operator, arXiv:1709.03634, 2017.