The paper contains the description of the optimal constant $\beta=3.4351\!\ldots$ for which the following inequality holds. Let $X$ be a real-valued martingale, $H$ be a predictable process taking values in $[-1,1]$ and let $Y$ be an Itô integral of $H$ with respect to $X$. Then \[ \Bigl\Vert\sup_{t\geq0}|Y_t|\Bigr\Vert_1\leq\beta\Bigl\Vert\sup_{t\geq0}|X_t|\Bigr\Vert_1. \] A version of this inequality in the discrete-time case is also established. The proof is based on Burkholder's technique, which relates the above estimate to the construction of an upper solution to a corresponding nonlinear three-dimensional problem.