The Hermite–Hadamard inequality was first considered for convex functions and has been studied extensively. Recently, many extensions were given with the use of general convex functions. In this paper we present some variants of the Hermite–Hadamard inequality for general convex functions in the context of q-calculus. From our theorems, we deduce some recent results in the topic.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Alp, N., Sarikaya, M.Z., Kunt, M., Iscan, I.: \(q\)-Hermite-Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. J. King Saud. Univ. Sci. 30, 193–203 (2018)
Bessenyei, M., Páles, Z.: On generalized higher-order convexity and Hermite-Hadamard-type inequalities. Acta Sci. Math. (Szeged) 70, 13–24 (2004)
S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite–Hadamard Inequalities, RGMIA Monographs, Victoria University (2000), available at rgmia.vu.edu.au/monographs/hermite_hadamard.html
T. Ernst, A Comprehensive Treatment of \(q\)-Calculus, Birkhäuser/Springer (Basel, 2012)
Jackson, F.H.: On \(q\)-functions and a certain difference operator. Trans. Roy. Soc. Edin. 46, 253–281 (1908)
Jackson, F.H.: On \(q\)-definite integrals. Quart. J. Pure Appl. Math. 41, 193–203 (1910)
Jhanthanam, S., Tariboon, J., Ntouyas, S.K., Nonlaopon, K.: On \(q\)-Hermite-Hadamard inequalities for differentiable convex functions. Mathematics 7, 632 (2019)
V. Kac and P. Cheung, Quantum Calculus, Springer (New York, 2002)
Klaričić, M., Neuman, E., Pečarić, J., Šimić, V.: Hermite-Hadamard's inequalities for multivariate \(g\)-convex functions. Math. Inequal. Appl. 8, 305–316 (2005)
Moslehian, M.S.: Matrix Hermite-Hadamard type inequalities. Houston J. Math. 39, 177–189 (2013)
Noor, M.A., Noor, K.I., Awan, M.U.: Hermite-Hadamard inequalities for modified \(h\)-convex functions. Transylv. J. Math. Mech. 6, 171–180 (2014)
Noor, M.A., Noor, K.I., Awan, M.U.: A new Hermite-Hadamard type inequality for \(h\)-convex functions. Creat. Math. Inform. 24, 191–197 (2015)
Sarikaya, M.Z., Saglam, A., Yildirim, H.: On some Hadamardtype inequalities for \(h\)-convex functions. J. Math. Inequal. 2, 335–341 (2008)
Tariboon, J., Ntouyas, S.K.: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Diff. Equ. 282, 1–19 (2013)
Tariboon, J., Ntouyas, S.K.: Quantum integral inequalities on finite intervals. J. Inequal. Appl. 2014, 121 (2014)
G. Toader, Some generalizations of the convexity, in: Proceedings of the Colloquium on Approximation and Optimization (Cluj-Napoca, 1985), Univ. of Cluj-Napoca (1985), pp. 329–338
Varošanec, S.: On \(h\)-convexity. J. Math. Anal. Appl. 326, 303–311 (2007)
The authors thank the anonymous referee for useful suggestions to improve the presentation of the results.
About this article
Cite this article
Bermudo, S., Kórus, P. & Nápoles Valdés, J.E. On q-Hermite–Hadamard inequalities for general convex functions. Acta Math. Hungar. (2020). https://doi.org/10.1007/s10474-020-01025-6
Key words and phrases
- Hermite–Hadamard inequality
- q-integral inequality
- h-convex function
- modified h-convex function
Mathematics Subject Classification