Suppose $X_t$, $0 \leq t \leq 1$ is a continuous positive super-martingale with $\mathbb{E}[X_0] = 1$. Then by Doob's maximal inequality we know that for any $\lambda > 0$, $$\mathbb{P}\left(\sup\limits_{0\leq t\leq 1}X_t > \lambda \right)\leq \frac{\mathbb{E}[X_0]}{\lambda} = \frac{1}{\lambda}.$$
I'm interested in the case that not only is $X_t$ a super-martingale, but also $\mathbb{E}[X_t]\leq 1-t$ for every $0 \leq t \leq 1$. In this case, can the above inequality be amplified to something like $$\mathbb{P}\left(\sup\limits_{0\leq t\leq 1}\frac{X_t}{f(t)} > \lambda \right) \leq \frac{1}{\lambda},$$ where $f(t)$ is some decreasing function. Ideally $f(t)$ could be something like $1-t$. Any references or suggestions are welcome.