EXTENSION OF SOME WELL-KNOWN POLYNOMIAL INEQUALITIES

Abstract

Let P(z) be a polynomial of degree n and P'(z) its derivative. In this paper we extend some well-known polynomial inequalities to operator B, which carries P into

 B[P(z)] = λ_0 P(z) +  λ_1 {nz} over 2 {P'(z)}over{1!} + λ_2 ({nz} over 2)^2 {P''(z)}over{2!}

where λ_0, λ_1 and λ_2 are real or complex numbers such that all the zeros of

 U(z) := λ_0 + C(n, 1)λ_1 z + C(n, 2)λ_2 z^2 ,  C(n, r) = {n!}over{r!(n-r)!},

lie in the half plane

 |z| ≤ |z - n over z|

and therefore obtain generalizations of these.