Hadamard-Hermite Integral Inequality: An Integration by Parts Point of View
Abstract
This article focuses on creating new integral inequalities based on, and derived from, the classical Hermite-Hadamard integral inequality for convex functions. A key aspect of these developments is the use of integration by parts. One of the results obtained provides an alternative perspective under a specific smoothness assumption.
Downloads
References
Beckenbach, E. F. (1948). Convex functions. Bulletin of the American Mathematical Society, 54(5), 439–460. https://doi.org/10.1090/S0002-9904-1948-08994-7
Bellman, R. (1961). On the approximation of curves by line segments using dynamic programming. Communications of the ACM, 4(6), 284. https://doi.org/10.1145/366573.366611
Budak, H., & Sarikaya, M. Z. (2021). On refinements of Hermite–Hadamard type inequalities with generalized fractional integral operators. Fractional Differential Calculus, 11(1), 121–132. https://doi.org/10.7153/fdc-2021-11-08
Chen, D., Anwar, M., Farid, G., & Bibi, W. (2023). Integral inequalities of Hermite–Hadamard type via q-h integrals. AIMS Mathematics, 8(7), 16165–16174. https://doi.org/10.3934/math.2023829
Chesneau, C. (2025). On several new integral convex theorems. Advances in Mathematical Sciences Journal, 14(4), 391–404. https://doi.org/10.37418/amsj.14.4.4
Chesneau, C. (2025). Examining new convex integral inequalities. Earthline Journal of Mathematical Sciences, 15(6), 1043–1049. https://doi.org/10.34198/ejms.15625.10431049
Chesneau, C. (2026). On two new theorems on convex integral inequalities. Advances in Mathematical Sciences Journal, 15(1), 67–75. https://doi.org/10.37418/amsj.15.1.5
Dragomir, S. S., & Agarwal, R. P. (1998). Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Applied Mathematics Letters, 11(5), 91–95. https://doi.org/10.1016/S0893-9659(98)00086-X
Hadamard, J. (1893). Étude sur les propriétés des fonctions entières et en particulier d'une fonction considérée par Riemann. Journal de Mathématiques Pures et Appliquées, 58, 171–215.
Hermite, C. (1883). Sur deux limites d'une intégrale définie. Mathesis, 3, 82.
Iddrisu, M. M., Okpoti, C. A., & Gbolagade, K. A. (2014). A proof of Jensen's inequality through a new Steffensen's inequality. Advances in Inequalities and Applications, 2014, 1–7.
Iddrisu, M. M., Okpoti, C. A., & Gbolagade, K. A. (2014). Geometrical proof of new Steffensen's inequality and applications. Advances in Inequalities and Applications, 2014, 1–10.
Jensen, J. L. W. V. (1905). Om konvekse Funktioner og uligheder mellem middelværdier. Nyt Tidsskrift for Matematik B, 16, 49–68.
Jensen, J. L. W. V. (1906). Sur les fonctions convexes et les inégalités entre les valeurs moyennes. Acta Mathematica, 30, 175–193. https://doi.org/10.1007/BF02418571
Mitrinović, D. S. (1970). Analytic inequalities. Springer-Verlag. https://doi.org/10.1007/978-3-642-99970-3
Mitrinović, D. S., Pečarić, J. E., & Fink, A. M. (1993). Classical and new inequalities in analysis. Kluwer Academic Publishers. https://doi.org/10.1007/978-94-017-1043-5
Niculescu, C. P. (2000). Convexity according to the geometric mean. Mathematical Inequalities & Applications, 3(2), 155–167. https://doi.org/10.7153/mia-03-15
Roberts, A. W., & Varberg, D. E. (1973). Convex functions. Academic Press.
Tariq, M., Ntouyas, S. K., & Shaikh, A. A. (2023). A comprehensive review of the Hermite–Hadamard inequality pertaining to fractional integral operators. Mathematics, 11(10), 1953. https://doi.org/10.3390/math11101953
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg){kind=logo}
