Zeros of an entire function

Question

If $f$ is an entire function with $|f(z)|>|f(\bar{z})|$ for all complex numbers $z$ in the upper half plane. How does this imply that $f$ has no zeros in the upper half plane?

Answer 1

The answer is simple: for each $z$ in the upper half plain $|f(z)|>|f(\bar z)|\geq 0$ hence $f(z)\neq 0$.