Bounds on Rudin-Shapiro polynomials of arbitrary degree

Let $P_{<n}(z)$ be the Rudin-Shapiro polynomial of degree $n-1$. We show that $|P_{<n}(z)|\le \sqrt{6n-2}-1$ for all $n\ge0$ and $|z|=1$, confirming a longstanding conjecture. This bound is sharp in the case when $n=(2\cdot 4^k+1)/3$ and $z=1$. We also show that for $n\ge m\ge0$, $|P_{<n}(z)-P_{<m}(z)|\le \sqrt{10(n-m)}$, which is asymptotically sharp in the sense that for any $\varepsilon>0$ there exists $n>m\ge0$ and $z$ with $|z|=1$ and $|P_{<n}(z)-P_{<m}(z)|\ge\sqrt{(10-\varepsilon)(n-m)}$, contradicting a conjecture of Montgomery.