ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 31, 30, 26 Tamrazov, Promarz M., Inst. of Math. of National Acad. of Sci. of Ukraine The solution of Eremenko’s extremal problem on harmonic functions Let: z be the complex variable; (u_{1}, u_{2}) be a couple of harmonic in |z| < 1 functions having the least harmonic majorant v and the greatest harmonic minorant w with v ≥ 0, w(0) ≥ 0; F be the class of all such couples; R := sup_{(u_{1},u_{2})∈F} max_{j=1,2} sup{r : r ≥ 0, u_{j}(z) ≥ 0 ∀z : |z| ≤ r}. A. Eremenko proved that 3 − 2\sqrt{2} ≤ R ≤ 2 − \sqrt{3}, posed the problem to find R and conjectured that R = 2 − \sqrt{3}. We have proved that R = 3−2\sqrt{2}, and established sharpend statements with complete description of all extremal cases for them. Furthermore, extentions are given for classes wider than R (without the requirement v ≥ 0), and also for collections of n functions harmonic in the unit ball of the Euclidean space R^{m} (∀n ≥ 2, m ≥ 2). (To be published in Complex Variables .)