I'd like to show that the following function is monotonically decreasing (from plotting it is obvious that it is):
$$f(n)=\frac{c\,(n^2-n)\, {}_2F_1(1,\, 2-n;\, n^2-2n+3;\, c)}{n^2-2n+2}$$
with domain the integers greater than 3 and with $0<c<1$. Mathematica gives me the derivative, but the expression looks rather complicated (again involving the hypergeometric function), and I can't tell what its sign is.
Can someone tell me how to proceed in this case?