Is there any characterization of the set of entire functions $f(z) $ such that $\Re(f(z)) \geq \Re(\overline{f(\bar{z})})$ for all $z\in \mathbb{C}^{+} $?

($\Re$ stands for the real part)

Edit: I was thinking in writing $f(z) $ in the form

$$f(z) =f(x+iy) =u(x, y) +iv(x, y) $$

and

$$\overline{f(\bar{z})}=\overline{f(x-iy)} =u(x, - y) - iv(x, - y) $$

so, it is like finding all such continuous functions $u$ for which $$u(x, y) \geq u(x, - y), \;\;\forall\; y>0, x\in\mathbb {R} $$