On minimal Hölder gaps and Shannon entropy balance (Q1656528)

Summary: 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 \(y\).