MathDB

Inequality on maximum value of polynomial in interval

Source:

July 12, 2012

inequalitiesalgebrapolynomialrotation

Problem Statement

Let

P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m

be a polynomial with complex coefficients such that

a_m\neq 0, a_n\neq 0

and

n>m

. Prove that

\text{max}_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^{n} |a_k|^2}