Inequalities for polynomials not vanishing in a disk

摘要

If P(z) is a polynomial of degree n, having no zeros in the unit disc, then for all α, β ∈ C with ∣α∣ ⩽ 1, ∣β∣ ⩽ 1, it is known that.P(Rz)-αP(z)+βR+12n-|α|P(z)⩽12Rn-α+βR+12n-|α||z|n+1-α+βR+12n-|α|max|z|=1|P(z)|,for R ⩾ 1 and ∣z∣ ⩾ 1.The present paper contains a generalization and an improvement of this and some other polynomial inequalities of similar nature.