# A simple proof of Popoviciu's inequality

## Authors

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

## Keywords:

Popoviciu's inequality, convex function, convex combination
Abstract views: 222

## Abstract

T. Popoviciu  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.

## References

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 DOI: 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.

2008-08-01

## How to Cite

Bencze, M., & Popovici, F. (2008). A simple proof of Popoviciu’s inequality. Rev. Anal. Numér. Théor. Approx., 37(2), 127–132. https://doi.org/10.33993/jnaat372-884

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