A friend recently told me the following two facts, for which he cannot recall a proof or a reference (but he remembers seeing them in the literature):

Let $f$ be a holomorphic function mapping the upper half-plane $H$ into itself. Let $G$ be the group of fractional linear transformations $(az+b)/(cz+d)$ where $ad-bc=1$, $a,d$ are odd integers and $b,c$ are even integers. Suppose that for every $g\in G$ and for every $z\in H$, $f(z)$ is not equal to $g(z)$. Then $f$ is fractional-linear. (Or maybe such $f$ just does not exist).

Let $f$ be the same as before. Suppose that for all integers $m,n$ and all $z\in H$ we have $f(z)\neq mz+n$. Then $f$ is fractional-linear.

I will appreciate any relevant reference or any other information.

EDIT: There is no $f\in Aut(H)$ that satisfies the condition of Problem 1. This implies that $f$ constructed by Aakumadula is NOT fractional-linear.

To prove this, we write $(az+b)/(cz+d)=(xz+y)/(uz+t)$, where $a,b,c,d$ are given real numbers, and we want to find integers $x,y,u,t$, where $x,t$ are odd, and $y,u$ are even, so that this has non-real roots $z$. This is to show that certain quadratic form in $a,b,c,d$ is indefinite. And this is performed by an elementary calculation.