MathDB

23

Part of 2024 China Team Selection Test

Problems(1)

Complex polynomial

Source: 2024 CTST P23

3/29/2024
P(z)=anzn+⋯+a1z+z0P(z)=a_nz^n+\dots+a_1z+z_0, with an≠0a_n\neq 0 is a polynomial with complex coefficients, such that when ∣z∣=1|z|=1, ∣P(z)∣≤1|P(z)|\leq 1. Prove that for any 0≤k≤n−10\leq k\leq n-1, ∣ak∣≤1−∣an∣2|a_k|\leq 1-|a_n|^2.
Proposed by Jingjun Han
polynomialalgebracomplex analysis2024 CTSTChina TST