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


Now, assume that:


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


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


for all $x, y$ in the domain of $f$.


(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?



A question on a special type of function
I'm assuming the first inequality needs absolute values also; if not then exchanging the roles of $x$ and $y$ forces a number and its opposite to both be greater than two.


In this case, there is such an $f$ which is injective:


Let $f(1)=1$.


Assume $f$ has been defined and satisfies the two properties on $\{1,\ldots,m\}$. To define $f(m+1)$ injectively, still satisfying the two above properties, we need that $f(m+1)$ is not in the range of $f$ as defined so far, and also a finite set of inequalities. ($\lfloor\sqrt{m}\rfloor$ of them if $m+1$ is not a square, and an additional $m$ of them if $m+1$ is a square.) If $f(m+1)$ is defined sufficiently large, all these conditions will be met.


Proceed by induction; the resulting function on $\mathbb{N}$ will be as desired.


You ask "What properties should $f$ posess?" If you're looking to contruct an $f$ which grows as slowly as possible, an inspection of the above construction could give one slowish example. 


