I am looking for a real, positive function $f$ satisfying $$\frac{f(m+1)+f(n)}{f(n+1)+f(m)}>\frac{n}{m}$$ with $n,m$ $\in \mathbb{N}$ and $n>m$. This inequality comes from physics and seems to be the condition to achieve bosonic condensation on a lattice.
Any ideas and suggestions welcome.
No, it's not possible. In fact, it's not possible to find a function $f:\mathbb N\to\mathbb R^+$ satisfying $$\frac{f(2)+f(n)}{f(1)+f(n+1)}>n$$ for all $n>1$ (this is your condition with $m=1$).
To see this, note that whatever positive values we take for $f(1)$ and $f(2)$, there will exist some $n_0$ such that $\frac{f(2)}{f(1)}<n_0$. Thus, for all $n\geq n_0$, in order to satisfy the condition we must have $$\frac{f(n)}{f(n+1)}>n_0.$$ (Here we use the fact that if $\frac{f(2)}{f(1)},\frac{f(n)}{f(n+1)}\leq n_0$ then $\frac{f(2)+f(n)}{f(1)+f(n+1)}\leq n_0$.)
Therefore, by repeated applicaton of the above, we have $f(n_0+k)<\frac{f(n_0)}{n_0^k}$, and so for some $n_1$ sufficiently large we have $f(n)<f(1)$ for all $n\geq n_1$. But now this means $$\frac{f(2)+f(n)}{f(1)+f(n+1)}<1+\frac{f(2)}{f(1)}$$ for all $n\geq n_1$, and this contradicts the required condition.