On the maximum modulus of a complex polynomial

In this paper we impose distinct restrictions on the moduli of zeros p(z)=n∑v=0avzvp(z)=∑v=0navzv and investigate dependence ∥p(Rz)−p(σz)∥‖p(Rz)−p(σz)‖, R>σ≥1R>σ≥1 MαMα Mα+πMα+π, where Mα=max1≤k≤n|p(ei(α+2kπ)/n)|Mα=max1≤k≤n|p(ei(α+2kπ)/n)| certain coefficients p(z)p(z). This comprises several results, which in particular yields some classical polynomial inequalities as special cases. Moreover, problem estimating p(1−wn)p(1−wn), $0<w\leq$ given $p(1)=0$ is considered.