Let $v\not= 1$ be a real number. Let $f(s)$ be real analytic on an open interval containing $v$ and $1$, with a zero of order $m\ge 1$ at $s=1$.

My question is: Can we solve this functional equation with respect to $f$:

$$f(sv)=\frac{(s-1)f'(s)}{f(s)}$$ where $s$ is in the proximity of $s=1$.