Given a Hilbert space valued martingale (Mₙ), let (M*ₙ) and (Sₙ(M)) denote its maximal function and square function, respectively. We prove that
𝔼|Mₙ| ≤ 2𝔼 Sₙ(M), n=0,1,2,...,
𝔼 M*ₙ ≤ 𝔼 |Mₙ| + 2𝔼 Sₙ(M), n=0,1,2,....
The first inequality is sharp, and it is strict in all nontrivial cases.