On Hollenbeck-Verbitsky conjecture for $4\mathrm{/}3<p<2$ and reverse Riesz-type inequalities for $2<p<4$

Let $P_{+}$ be the Riesz’s projection operator and let $P_{-}=I-P_{+}\mathrm{.}$ We find the best upper estimates of the expression ${\parallel {({\mid P_{+}f\mid }^s+{\mid P_{-}f\mid }^s)}^{1\mathrm{/}s}\parallel }_p$ in terms of Lebesgue p-norm of the function $f\in L^p(\mathbf{T})$ for $p\in (4\mathrm{/}3,2)$ and $0<s\le \frac{p}{p-1},$ thus extending results from \cite{Melentijevic_2022} and \cite{Melentijevic_2023}, where the mentioned range is not considered. Also, we find the best lower estimates of the same quantities for $p\in (2,4)$ and $s\ge \frac{p}{p-1},$ thus extending results from \cite{melentijevic-reverse-2025}.