Prove that if $f$ is analytic on $\Bbb C$, $f(0)=1/2$ and $|f(z)|\le|e^z-1/2|$ then $f(z)=e^z-1/2$

Question

I deduced that if $e^z-1/2=0$ then $f(z)=0$, but don't know how to proceed. Any advice is welcome.