Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture

Details Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture

2009-08-18

arxiv.org/abs/0908.2587v1

Mathematics

Complex Variables

Scientific paper

The goal of this paper is to prove the conjecture of Krzyz posed in 1968 that for nonvanishing holomorphic functions $f(z) = c_0 + c_1 z + ...$ in the unit disk with $|f(z)| \le 1$, we have the sharp bound $|c_n| \le 2/e$ for all $n \ge 1$, with equality only for the function $f(z) = \exp [(z^n - 1)/(z^n + 1)]$ 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 $n \le 5$. Our approach is completely different and relies on complex geometry and pluripotential features of convex domains in complex Banach spaces.

Affiliated with

Also associated with

No associations

Rating

Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-8827