BEST CONSTANT IN ZYGMUNDS INEQUALITY AND RELATED ESTIMATES FOR ORTHOGONAL HARMONIC FUNCTIONS AND MARTINGALES

Title & Authors
BEST CONSTANT IN ZYGMUNDS INEQUALITY AND RELATED ESTIMATES FOR ORTHOGONAL HARMONIC FUNCTIONS AND MARTINGALES

Abstract
For any $\small{K}$ > $\small{2/{\pi}}$ we determine the optimal constant $\small{L(K)}$ for which the following holds. If $\small{u}$, $\small{tilde{u}}$ are conjugate harmonic functions on the unit disc with $\tilde{u}(0) Keywords harmonic function;martingale;LlogL inequality;differential sub-ordination;best constants; Language English Cited by References 1. R. Banuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), no. 3, 575-600. 2. R. Banuelos and G. Wang, Orthogonal martingales under differential subordination and application to Riesz transforms, Illinois J. Math. 40 (1996), no. 4, 678-691. 3. R. Banuelos and G. Wang, Davis's inequality for orthogonal martingales under differential subordination, Michigan Math. J. 47 (2000), no. 1, 109-124. 4. D. L. Burkholder, Differential subordination of harmonic functions and martingales, Harmonic analysis and partial differential equations (El Escorial, 1987), 1-23, Lecture Notes in Math., 1384, Springer, Berlin, 1989. 5. C. Dellacherie and P. A. Meyer, Probabilities and Potential B, North-Holland, Amsterdam, 1982. 6. M. Essen, D. F. Shea, and C. S. Stanton, Best constants in Zygmund's inequality for conjugate functions, Papers on analysis, 73-80, Rep. Univ. Jyvaskyla Dep. Math. Stat., 83, University of Jyvaskila, 2001. 7. M. Essen, D. F. Shea, and C. S. Stanton, Sharp L$log^{\alpha}\$ L inequalities for conjugate functions, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 2, 623-659.

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