An Optimal Stability Theorem for H\"older's Inequality

We prove an optimal $L^1$ stability theorem for H\"older's inequality. Let $p>1$, $q>1$, and $1/p+1/q=1$. If $a_k,b_k\ge 0$ and \[ \sum_{k=1}^n a_k=\sum_{k=1}^n b_k=1, \] then \[ 1-\sum_{k=1}^n a_k^{1/p}b_k^{1/q} \ge \frac1{2pq}\left(\sum_{k=1}^n |a_k-b_k|\right)^2 . \] The constant $1/(2pq)$ is best possible. We also give the corresponding integral form.