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.
English version (Springer): Ukrainian Mathematical Journal 49 (1997), no. 8, pp 1276-1278.
Citation Example: Grigor'ev Yu. A. Some inequalities for gradients of harmonic functions // Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1135–1136.
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