Let $f:\mathbb{C} \to \mathbb{C}$ be a function of the complex variable $z$, satisfying the functional equation: $$\tag{1} z\cdot f(z)= \frac{z-2}{2} +(z-3)\cdot f(z-1).$$ It is easy to verify that $g(z)=\frac{3z-5}{24}$ is a solution to (1).
Are there other solutions?
More precisely, what condition -the least restrictive the better- on $f$ would be needed for $g$ be the only solution to (1)?