  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.
