Uniqueness of entire functions and fixed points

Let $f$ be a nonconstant entire function. %If $f(z)=z$ $\Longleftrightarrow $ $f'(z)=z$, and %$f'(z)=z$ $\Longrightarrow $ $f''(z)=z$, then $f\equiv f'$. In particular, If $f$, $f'$ and $f''$ have the same fixed points, then $f\equiv f'.$