**Stephen Montgomery-Smith and Shih-Chi Shen, An Extension to the Tangent Sequence Martingale Inequality.**
For each \(1 < p < \infty\), 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 \(E[s_k \vee d_k] \le E[s_k \vee e_k]\) then we have \({\|\sum d_k\|}_p \le c_p {\|\sum e_k\|}_p\).
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