The $L_q$ norm of the Rudin-Shapiro polynomials on subarcs of the unit circle

Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. Let $P_k$ and $Q_k$ denote the Rudin-Shapiro polynomials of degree $n-1$ with $n:=2^k$. For polynomials $S$ we define $$M_q(S,[\alpha ,\beta ]):={\left(\frac{1}{\beta -\alpha }{\int }_{\alpha }^{\beta }{\mid S(e^{it})\mid }^{q}dt\right)}^{1\mathrm{/}q},q>0\mathrm{.}$$ Let $\gamma :={\sin }^2(\pi \mathrm{/}8)$. We prove that $$\frac{\gamma }{4\pi }(\gamma n{)}^{q\mathrm{/}2}\le M_q(P_k,[\alpha ,\beta ]{)}^q\le (2n{)}^{q\mathrm{/}2}$$ for every $q>0$ and $32\pi \mathrm{/}n\le \beta -\alpha$. The same estimates hold for $P_k$ replaced by $Q_k$.