The Lei–Zhang Entropy Theorem and the Krzyż Conjecture

Recently, Jialin Lei and Tang Zhang proved that the entropy of a non-constant polynomial with all roots on the unit circle is maximized if the zeroes are equally spaced. Their result has an application to the Krzyż conjecture, which is that for non-vanishing holomorphic functions that map the disk to the disk, the maximum modulus of the \(n^{\mathrm{th}}\) Taylor coefficient is \(2/e\). I will discuss the Lei–Zhang theorem, and explain how it relates to the Krzyż conjecture.