Optimal interpolation inequalities between generalized means
Résumé
We exhibit the optimal inequalities of the form
\begin{equation*}
\begin{split}
x_1+\cdots+x_n\le\alpha (x_1^p+\cdots+x_n^p)^{1/p}+\beta x_1^{1/n}\cdots x_n^{1/n},\ \forall\, x_1,\ldots, x_n\ge 0,
\\
\text{with $n\ge 2$ and $1n$, a qualitative change occurs in the second inequality, and we present some partial results in the range $p>n\ge 3$.
As a byproduct of our analysis, we obtain a quantitative form of the strict convexity of the Shannon entropy.
Domaines
Mathématiques [math]| Origine | Fichiers produits par l'(les) auteur(s) |
|---|