Examining New Convex Integral Inequalities
Abstract
In this article, we introduce new convex integral inequalities based on a contemporary and adaptable analytical framework. These inequalities can handle composed functions, integral expressions, and ratio-type functionals, which make them applicable to a wide range of analysis problems. Our main result, in particular, complements a recent theorem from the literature by providing a valuable and non-trivial lower bound. The proofs are presented in full detail to ensure mathematical rigor, clarity, and reproducibility.
References
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.
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
Beckenbach, E. F. (1948). Convex functions. Bulletin of the American Mathematical Society, 54, 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
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.
Roberts, A. W., & Varberg, P. E. (1973). Convex functions. Academic Press.
Niculescu, C. P. (2000). Convexity according to the geometric mean. Mathematical Inequalities and Applications, 3(2), 155–167. https://doi.org/10.7153/mia-03-19
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.
Steffensen, J. F. (1918). On certain inequalities between mean values, and their application to actuarial problems. Skandinavisk Aktuarietidskrift, 1918, 82–97. https://doi.org/10.1080/03461238.1918.10405302
Bergh, J. (1973). A generalization of Steffensen's inequality. Journal of Mathematical Analysis and Applications, 41, 187–191. https://doi.org/10.1016/0022-247X(73)90193-5
Sulaiman, W. T. (2005). Some generalizations of Steffensen's inequality. Australian Journal of Mathematical Analysis, 2, 1–8.
Chesneau, C. (2025). New variants of the Steffensen integral inequality. Asian Journal of Mathematics and Applications, 2025(1), 1–12.
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