# On minimal Hölder gaps and Shannon entropy balance

### Immanuel M. Bomze

Universität Wien, Austria

## Abstract

When estimating a bilinear form in $x$ and $y$ by a product of two terms depending solely on $x$ or $y$, the well known Hölder inequality which uses the product of a $p$-norm and its dual comes easily into play. However, if one can choose $p$ freely, one could reduce this Hölder gap accordingly. This note addresses this elementary but apparently not too popular issue by using strict log-convexity of the $p$-norm in $\frac 1p$ (sometimes called Littlewood's inequality). The optimal $p$ is characterized by a balance condition on the Shannon entropies of distributions related to $x$ and $y$.

## Cite this article

Immanuel M. Bomze, On minimal Hölder gaps and Shannon entropy balance. Port. Math. 75 (2018), no. 1, pp. 1–10

DOI 10.4171/PM/2009