Maximal mean value of a differentiable function with small variation

Question

Let $B$ be a real valued function on $[0, 1]$ such that $\|B'\|_{L_2}\le 1$ and $B(0) = 0$. Denote \begin{gather}\label{small var condition} \varepsilon = \int_0^1 B^2(s)\, ds - \left(\int_0^1 B(s)\, ds\right)^2. \end{gather} Question. How large can the following quantities be in terms of $\varepsilon$: $$ \left|\int_0^1 B(s)\,ds\right|, \qquad \sup_{s\in[0,1]} |B(s)|? $$ I believe that using Cauchy-Schwarz inequality couple of times one can show the bounds $O(\sqrt[4]{\varepsilon})$ and $O(\sqrt[8]{\varepsilon})$ respectively. However I'm not sure if they are optimal.