  We state and discuss several interrelated results, conjectures, and questions
regarding contractive inequalities for classical $H^p$ spaces of the unit disc.
We study both coefficient estimates in terms of weighted $\ell^2$ sums and the
Riesz projection viewed as a map from $L^q$ to $H^p$ with $q\ge p$. Some
numerical evidence is given that supports our conjectures.
