Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture

2009/08/18 by Krushkal, Samuel L.
#30C50 #30C55 #32Q45 #Complex Variables (math.CV) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.0908.2587

Abstract

The goal of this paper is to prove the conjecture of Krzyz posed in 1968 that for nonvanishing holomorphic functions f(z) = c₀ + c₁ z + ... in the unit disk with |f(z)| ≤ 1, we have the sharp bound |c_n| ≤ 2/e for all n ≥ 1, with equality only for the function f(z) = exp [(zⁿ - 1)/(zⁿ + 1)] and its rotations. The problem was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for n ≤ 5. Our approach is completely different and relies on complex geometry and pluripotential features of convex domains in complex Banach spaces.

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