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Dear all,

the following is a special case of Stein inequalities for martingales.

$\textbf{Theorem}$ Let $(\Omega, \mathbb{P})$ be a (standard) probability space equipped with a filtration of increasing conditional expectations $\mathbb{E}_n$, $n \ge 1$. Then for any $ 1< p< \infty$, there exists a numerical constant $\gamma_p>0$ such that for any finite sequence of real valued functions $f_1, f_2, \cdots, f_n$ in $L_p(\Omega,\mathbb{P})$, we have $$\left\lVert \left(\sum_{k = 1}^n |\mathbb{E}_kf_k|^2\right)^{1/2}\right\rVert_p \le \gamma_p\left\lVert\left(\sum_{k = 1}^n |f_k|^2\right)^{1/2}\right\rVert_p.$$ ($\gamma_p$ is denoted to be the optimal constant in the above inequality.)

$\textbf{Question}:$ Is the following holds?

There exist $M> 0$, such that $$\gamma_p \ge 1 + M(p-2),$$ when $0 < p-2 <<1$.

$\textbf{Special case}:$ Let $C_q$ be the best constant verifying $$\| f_1^2 + \mathbb{E}(f_2)^2\|_q \le C_q \| f_1^2 + f_2^2\|_q.$$ Does there exist $M > 0$ such that $$ C_q \ge 1 + M(q-1)$$ for all $0 < q-1 \ll 1$.

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By the way, deducing from the duality of Doob's inequality, it is known that the converse inequality holds for some $M > 0$. –  Yanqi QIU Sep 3 '12 at 16:29
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