Let $0<t_0<1$ fixed number , $ n_0$ integer $ \geq 2$ fixed and let $\forall 0<u<1, f(u)= \displaystyle \frac{(1-u)^{n_0} \log(1-u)}{(1-ut_0)^{n_0+1}} $.

Let $0<u_0<1 $ be given. I'm looking a good upper bound ( wich is an explicit function in $u_0,t_0, n_0$) to $ |f^{(n_0)}(u_0)|$ ( derivative $n_0$ ieme) .

I suppose that by Cauchy formula, and choosing the good radius one can have the good upper bound. but i dont know how to do it. Any help?