Some inequalities for gradients of harmonic functions
For a function u(x, y) harmonic in the upper half-plane y>0 and represented by the Poisson integral of a function v(t) ∈ L 2 (−∞,∞), we prove that the inequality \(grad u (x, y)|^2 14π ^3 _ - ∞ ^∞ v^2 (t)dt\) is true. A similar inequality is obtained for a function harmonic in a disk., number=8, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Grigor'ev, Yu. A., year=1997, month=Aug., pages=1135–1136