Some inequalities for gradients of harmonic functions - Grigor'ev, Yu. A. (Ukrains’kyi Matematychnyi Zhurnal, 1997)
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 \(grad u (x, y)|^2 {\text{ }} \leqslant {\text{ }}\frac{1}{{4π ^3 }}{\text{ }}\int\limits_{ - ∞ }^∞ {v^2 } (t)dt\) is true. A similar inequality is obtained for a function harmonic in a disk.