Good-lambda bound for Brownian stopping time and maximum ($\mathbb{P}(\sqrt{\tau }>\beta \lambda ,B_{\tau }^{\ast }\le \delta \lambda )\le {\delta }^2({\beta }^2-1{)}^{-1}\mathbb{P}(\sqrt{\tau }>\lambda )$)

For Brownian motion started at zero, condition at time ${\lambda }^2$. If the stopping time exceeds ${\beta }^2{\lambda }^2$ while the running absolute maximum stays below $\delta \lambda$, the path must remain in a fixed interval for another $({\beta }^2-1){\lambda }^2$ time units. Its expected exit time from this interval is at most ${\delta }^2{\lambda }^2$. Markov's inequality yields the displayed bound. Integrating it against $d\Phi$ for a function of moderate growth, then absorbing the small tail term, proves $\mathbb{E}\Phi (\sqrt{\tau })\le C_{\Phi }\mathbb{E}\Phi (B_{\tau }^{\ast })$.