The Schwarzian derivative of an entire holomorphic function $f$ is defined as $$Sf:=\left(\frac{f^{''}}{f'}\right)'-\frac{1}{2}\left(\frac{f^{''}}{f'}\right)^2.$$

In the following, we only consider entire holomorphic functions.

If $Sf=0$, then it is well known that $f$ must be a linear function.

If $Sf$ is a constant, then it is also well known that $f$ takes the form $f(z)=a+be^{cz}$, where $a,b,c$ are complex numbers.

**My question** is the following: Let $f$ be a locally injective entire function (the injectiveness is to guarantee that $Sf$ is also entire), if $Sf$ is a polynomial of degree bigger than or equal to $1$, what does $f$ look like? Can anyone give an example?