Solution
= Solution
Fix any $z_0\in\mathbb C$. The <Cauchy estimate> on the circle $|z-z_0|=R$ applies for every $R>0$ because $f$ is an <entire function>. It gives
$$
|f'(z_0)|\leq\frac{\max_{|z-z_0|=R}|f(z)|}{R}\leq\frac{1+\sqrt{|z_0|+R}}R\longrightarrow0.
$$
Therefore $f'(z_0)=0$. Since $z_0$ was arbitrary and $\mathbb C$ is <connected>, \b[$f$ is constant]. This is the sublinear case of the <polynomial growth theorem for entire functions>.