Let f, g be two Hilbert-space-valued martingales such that g is differentially subordinate to f. The paper contains the proof of the estimate ‖g‖p,∞≤2p(p+1)p−1‖f‖p,∞,1p∞. The constant is shown to be of optimal order for p→∞ and for p→1. Related results for transforms of...