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

Let $f$ be a real-valued continuous function on the interval $[0,1]$ and satisfy the following estimate $$ \left|\int_0^1 f(t) e^{st}dt\right|\le Cs^{\frac12},\quad s>1, $$ where the constant $C$ is independent of $s$. Can we assert that $f$ is identically zero on $[0,1]$?

Let $f$ be a real-valued continuous function on the interval $[0,1]$ and satisfy 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 $[0,1]$?

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