An Optimal Stability Theorem for Hölder's Inequality

We prove an optimal $L^1$ stability theorem for Hölder’s inequality. Let $p>1$, $q>1$, and $1\mathrm{/}p+1\mathrm{/}q=1$. If $a_k,b_k\ge 0$ and $$\sum _{k=1}^n{a}_k=\sum _{k=1}^n{b}_k=1,$$ then $$1-\sum _{k=1}^n{a}_k^{1\mathrm{/}p}{b}_k^{1\mathrm{/}q}\ge \frac{1}{2pq}{(\sum _{k=1}^n\mathrm{\mid }{a}_k-b_k\mathrm{\mid })}^2\mathrm{.}$$ The constant $1\mathrm{/}(2pq)$ is best possible. We also give the corresponding integral form.