Hollenbeck-Verbitsky conjecture on best constant inequalities for analytic and co-analytic projections

\begin{abstract} In this paper we address the problem of finding the best constants in inequalities of the form: $$ \|\big(|P_+f|^s+|P_-f|^s\big)^{\frac{1}{s}}\|_{L^p({\mathbb{T}})}\leq A_{p,s} \|f\|_{L^p({\mathbb{T}})},$$ where $P_+f$ and $P_-f$ denote analytic and co-analytic projection of a complex-valued function $f \in L^p({\mathbb{T}}),$ for $p \geq 2$ and all $s>0$, thus proving Hollenbeck-Verbitsky conjecture from \cite{HV.OTAA}. We also prove the same inequalities for\\ $1