Best constants in reverse Riesz-type inequalities for analytic and co-analytic projections

Abstract: Let $P_+$ be the Riesz's projection operator and let $P_-= I - P_+$. We consider the inequalities of the following form $$ |f|{L^p(\mathbb{T})}\leq B{p,s}|( |P_ + f | ^s + |P_- f |^s) ^{\frac 1s}|{L^p (\mathbb{T})} $$ and prove them with sharp constant $B{p,s}$ for $s \in [p',+\infty)$ and $1<p\leq 2$ and $p\geq 4,$ where $p':=\min{p,\frac{p}{p-1}}.$