If $\abs{f}\le M$ on $\CC$, the Cauchy estimate on $\abs{z-z_0}=R$ gives $\abs{f'(z_0)}\le M/R$ for every $R>0$, so $f'=0$ and $f$ is constant.Suggest a slogan
Let \(f\) be an entire function. If there is \(M>0\) with \(\abs{f(z)}\leq M\) for all \(z\in\CC\), then \(f\) is constant.