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

\begin{abstract} In this paper we address the problem of finding best constants in inequalities 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 co-analytic projection a complex-valued function $f \in L^p({\mathbb{T}}),$ for $p \geq 2$ all $s>0$, thus proving Hollenbeck-Verbitsky conjecture from \cite{HV.OTAA}. We also prove same for\\ $1<p\leq\frac{4}{3}$ $s\leq \sec^2\frac{\pi}{2p}$ confirm that $s=\sec^2\frac{\pi}{2p}$ is sharp cutoff $s.$ The proof uses method plurisubharmonic minorants an approach appropriate "elementary" seems to be new topic. show result implies projections on real-line half-space multipliers $\mathbb{R}^n$ analog martingales. A remark isoperimetric inequality harmonic functions unit disk given. \end{abstract}