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)... (z(n) - z) with z(1)z(2)... z(n) = 1. Of interest here is the value [L] = sup{min{\L(p)(z(k))\ : k = 1, 2,..., n} : z(1)z(2)... z(n) = 1} 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(Sigma(n)(k=0) a(k)z(k)) = Sigma(n)(k=0) 1/k+1 a(k)z(k) = 1/z integral(z)(o) p(u) du, a(o)=1, a(n) = (-1)(n).