A simple proof of Popoviciu's inequality


  • Mihaly Bencze Aprily Lajos College, Braşov, Romania
  • Florin Popovici Grigore Moisil College, Braşov, Romania


Popoviciu's inequality, convex function, convex combination
Abstract views: 196


T. Popoviciu [5] has proved in 1965 the following inequality relating the values of a convex function \(f:I\rightarrow\mathbb{R}\) at the weighted arithmetic means of the subfamilies of a given family of points \(x_{1},...,x_{n}\in I\):\begin{align*}& \sum\limits_{1\leq i_{1}<\cdots <i_{p}\leq n}(\lambda_{i_{1}}+\cdots +\lambda _{i_{p}})\,f\left( \tfrac{\lambda_{i_{1}}x_{i_{1}}+\cdots +\lambda_{i_{p}}x_{i_{p}}}{\lambda _{i_{1}}+\cdots +\lambda _{i_{p}}}\right) \\& \leq \tbinom{n-2}{p-2}\left[\tfrac{n-p}{p-1}\,\sum\limits_{i=1}^{n}\,\lambda_{i}\,f(x_{i})+\left( \sum\limits_{i=1}^{n}\,\lambda _{i}\right) \,f\left( \tfrac{\lambda _{1}x_{1}+\cdots +\lambda _{n}x_{n}}{\lambda _{1}+\cdots+\lambda _{n}}\right) \right] .\end{align*}Here \(n\geq 3,\) \(p\in \{2,...,n-1\}\) and \(\lambda _{1},...,\lambda_{n}\) are positive numbers (representing weights). The aim of this paper is to give a simple argument based on mathematical induction and a majorization lemma.


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Bencze, M., Niculescu, C. P. and Popovici, F., Convexity according to Popoviciu's inequality, submitted.

Niculescu, C. P. and Persson, L.-E., Convex Functions and their applications. A Contemporary Approach, CMS Books in Mathematics, vol. 23, Springer-Verlag, New York, 2006, https://link.springer.com/book/10.1007%2F0-387-31077-0

Niculescu, C. P. and Popovici, F., A Refinement of Popoviciu's Inequality, Bull. Soc. Sci. Math. Roum., 49 (97), no. 3, pp. 285-290, 2006.

Pečarić, J. E., Proschan, F. and Tong, Y. C., Convex functions, Partial Orderings and Statistical Applications, Academic Press, New York, 1992, https://doi.org/10.1016/s0076-5392(08)x6162-4

Popoviciu, T., Sur certaines inégalités qui caractérisent les fonctions convexes, Analele Ştiinţifice Univ. "Al. I. Cuza", Iaşi, Secţia Mat., 11, pp. 155-164, 1965.




How to Cite

Bencze, M., & Popovici, F. (2008). A simple proof of Popoviciu’s inequality. Rev. Anal. Numér. Théor. Approx., 37(2), 127–132. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2008-vol37-no2-art4