Extremal problems for algebraic polynomials

Let $L(p)$ be a linear operator on the set of monic algebraic polynomials $p(z)= (z_1-z)(z_2-z)\dotsb(z_n-z)$ with $z_1z_2\dotsb z_n=1$. Of interest here is the value
$$ [L]=\sup\bigl\{\min\{|L(p)(z_k)|:k=1,2,\dots,n\}:z_1z_2\dotsb z_n=1\bigr\} $$
for various linear operators. The motivation is that Smale's mean value conjecture may be formulated as $[L]=1-1/(n+1)$ for the linear operator
$$ L(p)(z)=L\biggl(\sum_{k=0}^na_kz^k\biggr)=\sum_{k=0}^n\frac1{k+1}a_kz^k=\frac1z\int_0^zp(u)\,du, \enskip a_0=1, \ \ a_n=(-1)^n. $$