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**fareastmovement** Let$\displaystyle f : \mathbb{C}\rightarrow \mathbb{C} $ be an entire function such that $\displaystyle \lim_{z\rightarrow \infty} \frac{f(z)}{z}=0$

Show that $\displaystyle f$ is a constant.

$\displaystyle \noindent\rule{\textwidth}{0.5pt}$

I don't know if $\displaystyle \frac{f(z)}{z}$ is analytic at $\displaystyle z = 0$ , therefore I cannot use the Maximum Modulus Principle.On the other hand, I cannot prove that the maximum value exists, therefore I cannot use the Cauchy's Inequality.

Thank you.