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

Let $P_+$ be the Riesz's projection operator and let $P_-=I-P_+.$ We find the best upper estimates of the expression $\left\lVert \left( \left\lvert P_+f \right\rvert ^s + \left\lvert P_-f \right\rvert ^s \right) ^{1/s} \right\rVert _p$ in terms of Lebesgue p-norm of the function $f \in L^p(\mathbf{T})$ for $p \in (4/3,2)$ and $0 < s \leq \frac{p}{p-1},$ 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 $s \geq \frac{p}{p-1},$ thus extending results from \cite{melentijevic-reverse-2025}.