Some generalized Hermite-Hadamard type inequalities by using the harmonic convexity of differentiable mappings

Year 2019, Volume: 68 Issue: 2, 1543 - 1555, 01.08.2019

Muhammad Amer Latif , Sabir Hussain

https://doi.org/10.31801/cfsuasmas.422991

https://izlik.org/JA49RL55ZX

Abstract

In this paper, a general identity involving a di¤erentiable mapping is established. By using mathematical analysis, Hölder inequality and some auxiliary results, new generalized Hermite Hadamard type inequalities for differentiable harmonically-convex functions are established. It is expected that the results established in this paper contain previously established results asspecial cases.

Keywords

Hermite-Hadamard’s inequality , convex function , harmonic convex function

References

• Chen, F. X. and Wu, S. H., Some Hermite-Hadamard type inequalities for harmonically κ-convex functions, The Scientific World Journal, 2014 (2014), Article ID 279158.
• Chen, F. and Wu, S., Fejér and Hermite-Hadamard type inequalities for harmonically convex functions, J. Appl. Math. 2014 (2014), Article ID 386806, 6 pages.
• He, C. -Y., Wang, Y., Xi, B. -Y. and Qi, F., Hermite--Hadamard type inequalities for (α, m)-HA and strongly (α, m)-HA convex functions, J. Nonlinear Sci. Appl., 10 (2017) 205--214.
• Hölder, O., Über einen Mittelwerthssatz , Götting Nachr. (1889), 38-47.
• Hadamard, J., Étude sur les Propriétés des Fonctions Entières en Particulier d'une Fonction Considérée par Riemann. Journal de Mathématique Pures et Appliquées, 58, 171-215.
• Hermite, Ch., Sur deux limites d'une integrale define, Mathesis 3 (1883) 82.
• İscan, İ., Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe Journal of Mathematics and Statistics, 43 (6) (2014) 935-942.
• İscan, İ. and Wu, S., Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Applied Mathematics and Computation, 238 (2014) 237-244.
• Jensen, J. L. W. V., Sur les fonctions convexes et les inégalités entre les voleurs mogernmes, Acta. Math., 30 (1906), 175-193.
• Latif, M. A., Dragomir, S. S. and Momoniat, E., Fejér type inequalities for harmonically-convex functions with applications, Journal of Applied Analysis and Computation, (Accepted)
• Latif, M. A., Dragomir, S. S. and Momoniat, E., Some Fejér type inequalities for harmonically-convex functions with applications to special means, International Journal of Analysis and Applications, 13 (1) (2017) 1-14.
• Noor, M. A., Noor, K. I. and Iftikhar, S., Hermite-Hadamard inequalities for strongly harmonic convex functions, J. Inequal. Spec. Funct., 7 (2016) 99-113.
• Aslam Noor, M., Inayat Noor, K. and Iftikhar, S., Some Newton's type inequalities for harmonic convex functions, J. Adv. Math. Stud., 9 (1) (2016) 7-16.
• Awan, M. U., Aslam Noor, M. M., Mihai, V. and Inayat Noor, K., Inequalities via harmonic convex functions: conformable fractional calculus approach, J. Math. Inequal., 12 (1) (2018) 143-153.
• Stolz, O., Grundzüge der Differential und Integralrechnung, Leipzig, Vol. 1, (1893), 35--36.
• Wang, W., İscan, İ. and Zhou, H., Fractional integral inequalities of Hermite-Hadamard type for m-HH convex functions with applications, Advanced Studies in Contemporary Mathematics (Kyungshang), 26 (3) (2016) 501-512.
• Wang, W. and Qi, J., Some new estimates of Hermite-Hadamard inequalities for harmonically convex functions with applications, International Journal of Analysis and Applications, 13 (1) (2017) 15-21.
• Zhang, T. -Y. and Qi, Feng, Integral inequalities of Hermite-Hadamard type for m-AH convex functions, Turkish Journal of Analysis and Number Theory, 3 (2) (2014) 60-64.