On minimal Hölder gaps and Shannon entropy balance

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 $\frac{1}{p}$ (sometimes called Littlewood's inequality). The optimal $p$ is characterized by a balance condition on the Shannon entropies of distributions related to $x$ and $y$.