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Pré-Publication, Document De Travail Année : 2023

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.
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Dates et versions

hal-04043615 , version 1 (23-03-2023)

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  • HAL Id : hal-04043615 , version 1

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Petru Mironescu. Optimal interpolation inequalities between generalized means. 2023. ⟨hal-04043615⟩
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