On the expected diameter of planar Brownian motion

Known results show that the diameter $d_1$ of the trace of planar Brownian motion run for unit time satisfies $1.595\le \mathbb{E}{d}_1\le 2.507$. This note improves these bounds to $1.601\le \mathbb{E}{d}_1\le 2.355$. Simulations suggest that $\mathbb{E}{d}_1\approx 1.99$.