On Hollenbeck-Verbitsky conjecture for $4/3 < p < 2$

Let \(P_+\) be the Riesz's projection operator and let \(P_-=I-P_+.\) We find best estimates of the expression \( ( P_+f ^s + P_-f ^s ) 1/s _p \) in terms of Lebesgue p-norm of the function \(f ∈ L^p(T)\) for \(p ∈ (4/3,2)\) and \(0 < s ≤ p/p-1,\) thus extending results from {Melentijevic_2022} and {Melentijevic_2023}, where the mentioned range is not considered.