Some inequalities for gradients of harmonic functions

Abstract

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 $grad u (x, y)|^2 {\text{ }} \leqslant {\text{ }}\frac{1}{{4\pi ^3 }}{\text{ }}\int\limits_{ - \infty }^\infty {v^2 } (t)dt$ is true. A similar inequality is obtained for a function harmonic in a disk.