In Revuz and Yor's "Continuous Martingales and Brownian Motion" exercise 1.22 we are given $X$ is a non-negative self-similar process (defined by $(X_{\lambda t},t \ge 0) \stackrel{(d)}{=} (\lambda X_t, t \ge 0)$ for all $\lambda > 0$) and define $S_p := \sup_{s \ge 0} (X_s - s^p)$ and $X_t^* := \sup_{s \le t} X_s$.
The problem is to show there exists a constant $c_p$ such that $\mathbb{P}(c_p (X_1^*)^q \ge a) \le \mathbb{P}(S_p \ge a)$ where $q$ is the Holder conjugate exponent. My idea is that since $X$ is self-similar, so is $X^*$ and so we have
\begin{align*} S_p &= \sup_{s \ge 0} (X_s - s^p) = \sup_{s \ge 0} (X_s^* - s^p) \stackrel{(d)}{=} \sup_{s \ge 0} (s X_1^* - s^p). \end{align*}
We can solve for $s$ to maximize the last expression to get $s = \left( \frac{X_1^*}{p}\right)^{1/(p-1)}$ so it seems like we would have $S_p \stackrel{(d)}{=} \frac{p-1}{p} (X_1^*)^q$, but this would give an exact equality instead of an inequality. I'm thinking my mistake is somewhere in applying the self-similarity of $X^*$ in the definition of $S_p$, but I'm not sure exactly why it's wrong or what the correct approach is.