An Extension to the Tangent Sequence Martingale Inequality

For each 1 < p < infinity, there exists a positive constant c_p, depending only on p, such that the following holds. Let (d_k), (e_k) be real-valued martingale difference sequences. If for for all bounded nonnegative predictable sequences (s_k) and all positive integers k we have E[s_k vee |e_k|] le E[s_k vee |d_k|] then for all positive integers n we have || sum_{k=1}^n e_k ||p le c_p || \sum{k=1}^n d_k ||_p .