Hyperbolic geodesics, Krzyz's conjecture and beyond

2016/03/08 by Krushkal, Samuel L.
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.02668

Abstract

In 1968, Krzyz conjectured that for non-vanishing 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. This conjecture was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for n ≤ 5. We provide here two different proofs of this conjecture and its generalizations based on completely different ideas.

Related

• Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture
• A new look at Krzyz's conjecture
• The sharp estimates of all initial taylor coefficients in the Krzyz's problem
• Decrease of bounded holomorphic functions along discrete sets
• Hyperbolic metrics, homogeneous holomorphic functionals and Zalcman's conjecture