Let $\operatorname{abs}$ denote the absolute value (of a complex number) and $\operatorname{sgn}$ the signum function. Also $\operatorname{Re}$ means the real part of a complex number.
Im looking for real-entire functions $f(z)$ such that for $\operatorname{abs}(\operatorname{Re}(z))>\frac{1}{3}$ and $z = a + bi$:
$$ \operatorname{sgn}( \operatorname{Re}(f(a+bi))) = \operatorname{sgn}(a) $$
or stated differently:
$$ f(a + bi) = g + h i $$ $$ 0 < ga $$
I was inspired by $\operatorname{Si}(z)$.
I also wondered how fast these functions can grow in the direction of positive or negative infinity.
Ps: for those unaware : A real-entire functions is a function that is entire(analytic on the entire complex plane) and maps all reals to reals.