On the Generalized Hermite-Hadamard Inequalities Involving Beta Function

Year 2021, Volume: 9 Issue: 1, 112 - 118, 28.04.2021

Mehmet Zeki Sarıkaya , Fatih Ata

Abstract

In this paper, we establish new generalized fractional integral inequalities of Hermite-Hadamard type which cover the previously published result such as Riemann integral, Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral.

Keywords

Riemann-Liouville fractional integral, beta function, convex function

References

• [1] A. Akkurt, Z. Kacar, H. Yildirim, Generalized Fractional Integral Inequalities for Continuous Random Variables, Journal of Probability and Statistics 2015(2015), Article ID 958980.
• [2] A. Akkurt, M. E. Yildirim, H. Yildirim, On some integral inequalities for (k,h)-Riemann-Liouville fractional inte- gral, New Trends in Mathematical Sciences (NTMSCI) 4 (1), 138-146, 2016.
• [3] S. S. Dragomir and R.P. Agarwal, Two inequalities for di¤ erentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. lett., 11(5) (1998), 91-95.
• [4] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional di¤ erential equations, North-Holland Mathematics Studies, 204, Elsevier Sci. B.V., Amsterdam, 2006.
• [5] U. S. Kirmaci, Inequalities for di¤ erentiable mappings and applications to special means of real numbers and to midpoint formula, Applied Mathematics and Computation 147 (2004) 137–-46
• [6] R. Goren‡o and F. Mainardi, Fractional calculus: integral and di¤ erential equations of fractional order, Springer Verlag, Wien (1997), 223-276.
• [7] J. Hadamard, Etude sur les proprietes des fonctions entieres et en particulier d’une fonction considree par, Rie- mann, J. Math. Pures. et Appl. 58 (1893), 171-215.
• [8] S. Miller and B. Ross, An introduction to the fractional calculus and fractional di¤ erential equations, John Wiley & Sons, USA, 1993, p.2.
• [9] R. K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators, East Asian Math. J., 21(2) (2005), 191-203.
• [10] M. Z. Sarikaya, E. Set, H. Yaldiz and N. Basak, Hermite-Hadamard 0s inequalities for fractional integrals and related fractional inequalities, Mathematical and Computer Modelling, 57 (2013) 2403-2407.
• [11] M. Z. Sarikaya and H. Yaldiz, On generalization integral inequalities for fractional integrals, Nihonkai Math. J., Vol.25(2014), 93-104.
• [12] M. Z. Sarikaya, H. Yaldiz and N. Basak, New fractional inequalities of Ostrowski-Grüss type, Le Matematiche, Vol. LXIX (2014)-Fasc. I, pp. 227-235.
• [13] M. Z. Sarikaya and F. Ertu¼gral, On the generalized Hermite-Hadamard inequalities, Accepted in Annals of the University of Craiova - Mathematics and Computer Science Series), 2019.
• [14] M. E. Ozdemir, S.S. Dragomir and C. Yildiz, The Hadamard’s inequality for convex function via fractional integrals, Acta Mathematica Scientia 2013,33B(5):1293-1299.
• [15] T. Ali, M. A. Khan and Y. Khurshidi, Hermite-Hadamard inequality for fractional integrals via eta-convex functions, Acta Mathematica Universitatis Comenianae, 86(1), (2017), 153-164.
• [16] M. Kunt and ·I. ·I¸scan, Hermite-Hadamard-Fejér type inequalities for p-convex functions. Arab J. Math. Sci. 23 (2017), no. 2, 215-230.
• [17] M. Kunt and ·I. ·I¸scan, Hermite-Hadamard type inequalities for harmonically ( , m)-convex functions by using fractional integrals, Konuralp J. Math. 5 (2017), no. 1, 201-213.
• [18] M. Kirane, B. T. Torebek, Hermite-Hadamard, Hermite-Hadamard-Fejer, Dragomir-Agarwal and Pachpatte Type Inequalities for Convex Functions via Fractional Integrals, arXiv:1701.00092.
• [19] M. Iqbal, M. I. Bhatti, and K. Nazeer, Generalization of Inequalities Analogous to Hermite–Hadamard Inequality via Fractional Integrals, Bull. Korean Math. Soc, 52 (2015), No. 3, 707-716.
• [20] S. Mubeen and G. M Habibullah, k-Fractional integrals and application, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 2, 89 - 94.
• [21] G. Farid, A. Rehman and M. Zahra, On Hadamard inequalities for k-fractional integrals, Nonlinear Functional Analysis and Applications Vol. 21, No. 3 (2016), pp. 463-478.
• [22] R. Hussain, A. Ali, G. Gulshan, A. Latif and M. Muddassar, Generalized co-ordinated integral inequalities for convex functions by way of k -fractional derivatives, Miskolc Mathematical Notesa Publications of the university of Miskolc. (Submitted)
• [23] R. Hussain, A. Ali, A. Latif, G. Gulshan, Some k–fractional associates of Hermite–Hadamard’s inequality for quasi–convex functions and applications to special means, Fractional Di¤erential Calculus, Volume 7, Number 2 (2017), 301-309.