(untitled) (Analysis in Theory and Applications, 2012)
Let p(z) be a polynomial of degree at most n. In this paper we obtain some new results about the dependence of\|p(Rz)−β p(rz)+α{R+1r+1)^n-|β|} p(rz)\|_s on \|p(z)\|_s for every α, β C with |α| ≤ 1, |β| ≤ 1, R > r 1, and s > 0. Our results not only generalize some well known inequalities, but also are variety of interesting results deduced from them by a fairly uniform procedure.