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For any > we determine the optimal constant for which the following holds. If , are conjugate harmonic functions on the unit disc with , then We also establish a related estimate for orthogonal harmonic functions given on Euclidean domains as well as an extension concerning orthogonal martingales under differential subordination.

BEST CONSTANT IN ZYGMUND'S INEQUALITY AND RELATED ESTIMATES FOR ORTHOGONAL HARMONIC FUNCTIONS AND MARTINGALES