Let $(B_t)_{t \geq 0}$ be a standard Brownian motion and let $T>0$ be fixed. I am interested in finding upper bounds for the expectations of the following running suprema: $$ \sup_{0 \leq t \leq T} |B_t|, \quad \sup_{0 \leq t \leq T} |B_t|^2. $$ We can use Doob's maximal inequality for the second term to get $$ \mathbb{E}\left(\sup_{0 \leq t \leq T} |B_t|^2\right) \leq 4 \sup_{0 \leq t \leq T} \underbrace{\mathbb{E}\left(|B_t|^2\right)}_{=t} = 4T. $$ Is there a straightforward way to bound $$ \mathbb{E}\left(\sup_{0 \leq t \leq T} |B_t| \right) \quad ? $$ Maybe the estimate $$ \mathbb{P}\left(\sup_{0 \leq t \leq T} |B_t| \geq x \right) \leq 2\mathbb{P}(|B_T| \geq x) $$ could be helpful.
EDIT: Some further progress
Using the estimate above and the fact that the running supremum under question is non-negative, we have $$ \mathbb{E}\left(\sup_{0 \leq t \leq T} |B_t| \right) = \int_0^{\infty} \mathbb{P}\left(\sup_{0 \leq t \leq T} |B_t| \geq x \right) dx \leq 2 \int_0^\infty \mathbb{P}(|B_T| \geq x) d x $$
