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 $$\| (\sum_{k = 1}^n |\mathbb{E}_kf_k|^2)^{1/2}\|_p \le \gamma_p\| (\sum_{k = 1}^n |f_k|^2)^{1/2}\|_p.$$ ($\gamma_p$ 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 $0 < p-2 <<1$.