IMPROVED HERMITE-HADAMARD TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA KATUGAMPOLA FRACTIONAL INTEGRALS

Year 2019, Volume: 37 Issue: 2, 461 - 474, 01.06.2019

Zeynep Şanlı Mehmet Kunt Tuncay Köroğlu

Abstract

In this paper, we gave adjustments for some results in the paper [1], and proved three new Katugampola fractional Hermite-Hadamard type inequalities for convex functions by using the left and the right fractional integrals independently. One of our Katugampola fractional Hermite-Hadamard type inequalities is better than given by Chen and Katugampola. Also, we gave two new Katugampola fractional identities for differentiable functions. By using these identities, we obtained some new trapezoidal type inequalities for convex functions. Our results generalize earlier results.

Keywords

Hermite-Hadamard inequalities, Katugampola fractional integrals, convex functions.

References

• [1] Chen, H., Katugampola, U. N., Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for generalized fractional integrals. J. Math. Anal. Appl. 446 (2017), 1274-1291.
• [2] Dragomir, S. S., Agarwal, R. P., Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91-95.
• [3] Hadamard, J., Etude sur les proprietes des fonctions entieres et en particulier d'une fonction consideree par Riemann, J. Math. Pures Appl., 58 (1893), 171-215.
• [4] Hermite, Ch., Sur deux limites d'une integrale definie, Mathesis, 3 (1883), 82-83.
• [5] Katugampola, U. N., New approach to generalized fractional integral, Appl. Math. Comput. 218 (3) (2011) 860-865.
• [6] Katugampola, U. N., Mellin transforms of generalized fractional integrals and derivatives, Appl. Math. Comput. 257 (2015) 566-580.
• [7] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and applications of fractional differential equations. Elsevier, Amsterdam (2006).
• [8] Kunt, M., Karapınar, D., Turhan, S., İşcan, İ., The left Rieaman-Liouville fractional Hermite-Hadamard type inequalities for convex functions, Mathematica Slovaca (2019) (Accepted for publication).
• [9] Kunt, M., Karapınar, D., Turhan, S., İşcan, İ., The right Rieaman-Liouville fractional Hermite-Hadamard type inequalities for convex functions, J. Inequal. Special Func., 9(1) (2018), 45-57.
• [10] Kunt, M., İşcan, İ., Turhan, S., Karapınar, D., Improvement of Fractional Hermite-Hadamard Type Inequality and Some New Fractional Midpoint Type Inequalities for Convex Functions, Miskolc Mathematical Notes (2019) (Accepted for publication).
• [11] Pearce, C. E. M., Pecaric, J., Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13 (2000), 51-55.
• [12] Roberts, A. W., Varberg, D. E., Convex functions, Academic Press, New York (1973.)
• [13] Set, E., Karaoğlan, A., Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for (k,h)-convex function via Katugampola fractional integrals, KJM, (5) (2017), 181-191.
• [14] Sarikaya, M. Z., Set, E. , Yaldız, H., Başak, N., Hermite-Hadamard's inequalities for fractionall integrals and related fractional inequalities, Math. Comput. Modelling 57(9) (2013) 2403-2407.