Let $f(z)$ be an entire function with $Re(f(z)) \geq 0$ for all $z$ in $\Bbb C$. Here if we consider $(\exp(z))(-f(z))$ {composition of two entire functions} is entire. Let $g(z) = e^{(-f(z))}$, then maximum magnitude of $g(z)$ turns to be $1$ as $Re(f(z))\geq 0$ and so maximum value of $\exp(Re(-f(z)))$ will be $1$ at $0$. So we get $g$ as entire and bounded function, hence constant. And therefore on solving we get $f$ is also constant.
Now if I consider $Re(f(z))=mod(z)$, for all $z$ in $\Bbb C$
for different value we give, we get different value for $Re(f(z))$. So $f(z)$ cannot be constant.
But from the argument I made above, function with real part $\geq 0$ must be constant. So here $Re(f(z))=mod(z)$ is positive. Should I consider whether $Re(f(z))$ is harmonic or not. Or ? I'm not able to figure out where I'm going wrong.
Any help would be much thankful.