Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $\{l_1,\dots,l_n\}$ a vector of natural numbers such that $l_1+l_2+\dots+l_n=N$. Let $ h_j(x)=\prod_{i\neq j,i=1,\dots, n} (x-x_i)^{-l_i}. $
Assume $|x_i|\leq 1$ for all $i=1,\dots,n$ and $|x_i-x_j|\geq \delta>0$ for all $i\neq j$.
Question: bound (from above) the successive derivatives of $h_j$ at $x_j$, i.e. $ |h_j^{(t)}(x_j)|,\qquad t=0,\dots,l_j-1 $ in terms of $N,n,\delta,t$.
(For motivation, see this MO question).