Maximum of modulus of a series involving polynomials

Question

Let $P(z)=\sum_{k=0}^na_kz^k $ be a polynomial of degree $n$ having no zeros in $|z|<1.$ Then what is the best value for 'M' in $$\frac{1+|a_0|}{n|a_0|}\left|\sum_{k=1}^nP(zw_k)\frac{w_k}{(w_k-1)^2}\right|\leq M$$ on $|z|=1$ where $w_k, 1\leq k\leq n$ are the roots of $z^n+|a_0|=0$