Hunt for a function. I am looking for any nontrivial function $f(z): \mathbb{C}\to\mathbb{C}$ such that:


*

*$f(z)$ is an entire function.

*A $z_p\in\mathbb{C}$ exists for which $\Re(f(z))\geq\Re(f(z_p))~\forall z\in\mathbb{C}$ with $\infty>\Re(f(z_p))>-\infty$.
If you do not have an example, but can give me some strategy on how to look for such a function, please share your thoughts. Thanks!
EDIT:
Ok, turns out the condition entire is too restrictive. Any idea by how much I should relax the restrictions to get the second property (and preferably the same for the imaginary part at the same $z_p$)?

Such a function must be constant.
Consider $g(z) = \exp(-f(z))$. We have
$$
\left|g(z)\right| = \exp(-\Re(f(z)) \le \exp(-\Re(f(z_p)).
$$
Since $g$ is entire and bounded, it must be constant. Hence $f$ is constant too.
