I am not sure if "recurrence inequality" is the correct term or whether it is possible to actually find an answer to this problem but anyways.
Let $n$ be a fixed natural number. Let $R(x,y)$ be a symmetric function of two variables (the domain is the set of whole numbers), i.e. $R(x,y)=R(y,x)$ and further suppose that:
Clearly $R(m,m)\le cn$ where $c$ is a natural number depending upon $m$. Is it possible to find the value of $c$ in terms of $m$?
I found that $R(1,1)\le 2n$ ; $R(2,2) \le 6n$ ; $R(3,3) \le 20n$ and $R(4,4) \le 70n$ by hand-computation. The pattern $2,6,20,70$ doesn't seem to be leading anywhere for me though. Does someone have an opinion concerning this?