MathDB
complex rational function

Source: VJIMC 2011 2.3

June 1, 2021
complex analysisalgebrarational functionfunction

Problem Statement

Let pp and qq be complex polynomials with deg⁡p>deg⁡q\deg p>\deg q and let f(z)=p(z)q(z)f(z)=\frac{p(z)}{q(z)}. Suppose that all roots of pp lie inside the unit circle ∣z∣=1|z|=1 and that all roots of qq lie outside the unit circle. Prove that max⁡∣z∣=1∣f′(z)∣>deg⁡p−deg⁡q2max⁡∣z∣=1∣f(z)∣.\max_{|z|=1}|f'(z)|>\frac{\deg p-\deg q}2\max_{|z|=1}|f(z)|.
Back to ProblemsView on AoPS