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

Let $P_{+}$ be the Riesz’s projection operator and let $P_{-}=I-P_{+}$. We consider the inequalities of the following form $$\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mathbb{T})}\le B_{p,s}\mathrm{\parallel }(\mathrm{\mid }{P}_{+}f{\mathrm{\mid }}^s+\mathrm{\mid }{P}_{-}f{\mathrm{\mid }}^s{)}^{\frac{1}{s}}{\mathrm{\parallel }}_{L^p(\mathbb{T})}$$ and prove them with sharp constant $B_{p,s}$ for $s\in [p^{\mathrm{\prime }},+\mathrm{\infty })$ and $1<p\le 2$ and $p\ge 4,$ where $p^{\mathrm{\prime }}:=\min \{p,\frac{p}{p-1}\}\mathrm{.}$