Wirtinger-Beesack integral inequalities.
We study the law of coexistence of different types of cycles for a continuous map of the interval. For this we introduce the notion of eccentricity of a pattern and characterize those patterns with a given eccentricity that are simplest from the point of view of the forcing relation. We call these patterns X-minimal. We obtain a generalization of Sharkovskiĭ's Theorem where the notion of period is replaced by the notion of eccentricity.
For an odd prime, we show that the Fekete polynomial has zeros on the unit circle, where . Here is the probability that the function has a zero in , where each is with y . In fact has absolute value at each primitive th root of unity, and we show that if for some then there is a zero of close to this arc.