Suppose I have a function $f$, and positive integers $x$ and $y$ such that $x$ is a
square and $x \ne y$.

Now, assume that:

$\frac{f(x)}{y} - \frac{f(y)}{x} > 2$

$|\frac{f(\sqrt{x})}{y} - \frac{f(y)}{\sqrt{x}}| > 1$

(Note that $|N|$ is the absolute value of $N$.)

My question is: What properties would this function $f$ necessarily possess?  In particular, can $f$ be injective?