For brevity's sake, define the following sequence of (Gaussian) hypergeometric functions for each $n\in\mathbb{N}$: $$f_n(z)={}_2 F_1(-n,-(n-1);2;z).$$ I wish to show that the logarithmic derivatives of these $f_n$'s satisfy these inequalities: $$\frac{d}{dz}\log(f_n(z))=\frac{f'_n(z)}{f_n(z)}\geq \frac{n}{z+\sqrt{z}}-\frac1{z},$$ for all $n$ and for $z\in(0,1)$.
Here are a few facts about these $f_n$ functions that may be useful. First, each $f_n$ is actually an $(n-1)$-degree polynomial, since the rising factorials of $-(n-1)$ in the coefficients eventually vanish. Secondly, it can be shown that $$f'_n(z)=\frac{n(n-1)}{2}{}_2F_1(-n+1,-(n-1)+1;3;z).$$
Any help with this problem that you can provide will be appreciated.