$f(z)$ is a entire function,and $\displaystyle{\lim\limits_{z\to \infty}\frac{Re(f(z))}{z}=0}$,then f(z) is a constant.
Actully I don't very understand the uasage of $Re(f(z))$,if there is someone told me about this. I will be very appreciate.
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$Re(f(z))$ is the real part of $f(z)$. I.e., if $f(z) = u(z) + iv(z)$, then $Re (f(z)) = u(z)$. – Lukas Geyer Apr 26 '17 at 01:13
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By the way, it follows from this more general result: https://math.stackexchange.com/questions/1647212/can-the-real-part-of-an-entire-function-be-bounded-above-by-a-polynomial – Lukas Geyer Apr 26 '17 at 01:18