Solving functional equation $\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)}$

Question

I want to find a functional equation $f(s,x)$ such that $$\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)}$$ If it helps the context I need this in is where $t$ is a member of a set of real number and $m$ is the maximum value of members that set. s > 0. Not being a mathematician by background i'm not familiar with appproches to solving this and Wolfram has not provided an answer.

Answer 1

Don't think this has any solutions as a functional equation.

From $\frac{f(s, t)}{f(s, m)} = \frac{1}{1+s(m-t)} $, if $m=t+1$, this gives $\frac{f(s, m+1)}{f(s, m)} = \frac{1}{1+s} $.

Applying this twice, $\frac{f(s, m+2)}{f(s, m)} = \frac{1}{(1+s)^2} $.

However, setting $m = t+2$ gives $\frac{f(s, m+2)}{f(s, m)} = \frac{1}{1+2s} $.

This is a contradiction.