Proof of the Agler--McCarthy entropy conjecture

In 2021, J.~Agler and J.~E. McCarthy proposed a two-step programme toward the celebrated Krzyż conjecture. The first step is to prove an entropy conjecture for polynomials whose zeros all lie on the unit circle; the second is to establish a full degree condition for extremal functions in the Krzyż conjecture. The purpose of this paper is to complete the first step. More precisely, we establish the sharp homogeneous entropy inequality for all non-constant polynomials with zeros on the unit circle and determine the equality cases.