Dear all,
we have by Stein that for any sequence $0 \le r_k \le 1$, and any functions $f_1, \cdots, f_n$ which are holomorphic on a neighbourhood of the unit disk, $$\| (\sum_{k =1}^n |f_k(r_k e^{i 2 \pi \theta})|^2)^{1/2}\|_{L_p([0, 1])} \le c_p\| (\sum_{k =1}^n |f_k(e^{i 2 \pi \theta})|^2)^{1/2}\|_{L_p([0, 1])}. $$
Let us assume that $c_p$ is the best constant appear in the above inequality.Of course, $c_2 =1$.
$\textbf{Question}$: What is the order $c_p$ when $p \to 2$. Could we have $$c_p \ge 1 + M(p-2)$$ for some fixed $M > 0$ when $p = 2 + \varepsilon$ for $\varepsilon$ very small?
