On Borwein's identity and weighted Turán type inequalities on a closed interval

Let $\Pi_n^*$ be the class of algebraic polynomials $P$ of degree $n$ having all zeros on the interval $[-1,1]$ and vanishing at the points $1$ and $-1$. In addition, let $w(x)=1-x^2$. The main result of the paper can be formulated as follows: there is an absolute constant $A>0$ such that
$$ \|P'w^{1-s}\|_{C[-1,1]}>A\sqrt{n}\cdot \sqrt{1-\Delta_P^2}\,\|Pw^{-s}\|_{C[-1,1]} $$
for any $P\in \Pi_n^*$ and $s\in [0,1]$, where $\Delta_P=\inf\big\{d\ge 0\colon \|Pw^{-s}\|_{C[-d,d]}=\|Pw^{-s}\|_{C[-1,1]}\big\}$. This inequality may be interpreted as a weighted analog of P. Turán's classical inequality for the derivative of polynomials with zeros on a closed interval. The proof uses a generalization of an interesting formula of P. Borwein concerning the logarithmic derivative of such polynomials. Our estimate is sharp in the order of the quantity $n$ and complements well-known results of V. F. Babenko, S. A. Pichugov, S. P. Zhou, and others.