My question is:

Let $f$ be a continous function on the interval $[0,1]$ and satisfies the following estimate $$ \left|\int_0^1 f(t) e^{st}dt\right|\le Cs,\quad s>1 $$ holds true, where the constant $C$ is independent of $s$. Can we assert $f$ is identically zero on $[0,1]$?

Thank you.