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. Let X be a Hilbert-space valued martingale and Ya real-valued supermartingale which are orthogonal and with Ydifferentially subordinate to X . ThenP fjY j  g KjjX jj 1 ;withK =1 +13 2 +15 2 +17 2 +19 2 + \Delta \Delta \Delta1 \Gamma132 +152 \Gamma172 +192 \Gamma \Delta \Delta \Delta=28fi(2) 1:34688525where the constant fi(2) is Catalan's constant whose approximatevalue is 0:915965594. The constant K is B. Davis" constant in the...

Davis's inequality for orthogonal martingales under differential subordination.