Logarithmic estimates for nonsymmetric martingale transforms

Let d=(d0,d1,d2,…) be a martingale difference sequence and θ=(θ0,θ1,θ2,…) be a predictable sequence taking values in [0,1]. In this paper we study the inequality supnE|∑k=0nθkdk|≤KsupnE|∑k=0ndk|log|∑k=0ndk|+L(K) and show that it holds with some universal L(K)<∞ if and only if K>1/2. Furthermore, we determine the optimal value of L(K) for K≥1 and the optimal order of L(K) as K→1/2. Related estimates for stochastic integrals are also established.