Corollary from Khintchine's inequality Let $z_1,\dots,z_n\in\mathbb{C}$ and $\epsilon_j\in\{-1,1\}$ for $j=1,\dots,n$ independent random variables with $P(\epsilon_j=\pm 1)=1/2$.  Khintchine's inequality states that 
$$A^{-1}\left(\sum_{j=1}^n |z_j|^2\right)^{1/2}\le\left(\mathbb{E}\left|\sum_{j=1}^n \epsilon_j z_j\right|^p\right)^{1/p}\le A\left(\sum_{j=1}^n |z_j|^2\right)^{1/2}$$
for some uniform constant $A>1.$
Now my question is: how can Khintchine's inequality be used to prove the following:
Let $T:L^p\rightarrow L^p$ be a bounded linear operator where $L^p=L^p(\mathbb{R}^d)$ and $1\le p<\infty$. Then
$$\left\|\left(\sum_{j=1}^n |Tf_j|^2\right)^{1/2}\right\|_p\le C\left\|\left(\sum_{j=1}^n |f_j|^2\right)^{1/2}\right\|_p$$
for $f_1,\dots,f_n\in L^p$ and $C>0$ a constant.
?
 A: Let $\{f_j\}_{j=1}^n\subset L_p(\Omega,\mu)$, then using Khinchine's inequality and Fubini's theorem we get
$$
\begin{align}
A^{-1}\left\Vert\left(\sum\limits_{j=1}^n|f_j|^2\right)^{1/2}\right\Vert_p
&\leq\left(\int_\Omega\mathbb{E}\left|\sum\limits_{j=1}^n\varepsilon_j f_j(\omega)\right|^pd\mu(\omega)\right)^{1/p}\\
&=\left(\mathbb{E}\left[\int_\Omega\left|\sum\limits_{j=1}^n\varepsilon_j f_j(\omega)\right|^pd\mu(\omega)\right]\right)^{1/p}\\
&=\left(\mathbb{E}\left[\left\Vert\sum\limits_{j=1}^n \varepsilon_j f_j\right\Vert_p^p\right]\right)^{1/p}
\end{align}
$$
Similarly one can show
$$
\left(\mathbb{E}\left[\left\Vert\sum\limits_{j=1}^n \varepsilon_j f_j\right\Vert_p^p\right]\right)^{1/p}
\leq A\left\Vert\left(\sum\limits_{j=1}^n|f_j|^2\right)^{1/2}\right\Vert_p
$$
hence
$$
A^{-1}\left\Vert\left(\sum\limits_{j=1}^n|f_j|^2\right)^{1/2}\right\Vert_p
\leq
\left(\mathbb{E}\left[\left\Vert\sum\limits_{j=1}^n \varepsilon_j f_j\right\Vert_p^p\right]\right)^{1/p}
\leq
A\left\Vert\left(\sum\limits_{j=1}^n|f_j|^2\right)^{1/2}\right\Vert_p
$$
Now we can prove the desired inequality
$$
\begin{align}
\left\Vert\left(\sum\limits_{j=1}^n|T(f_j)|^2\right)^{1/2}\right\Vert_p
&\leq
A\left(\mathbb{E}\left[\left\Vert\sum\limits_{j=1}^n\varepsilon_j T(f_j)\right\Vert_p^p\right]\right)^{1/p}\\
&\leq
A\left(\mathbb{E}\left[\left\Vert T\left(\sum\limits_{j=1}^n\varepsilon_j f_j\right)\right\Vert_p^p\right]\right)^{1/p}\\
&\leq
A\left(\mathbb{E}\left[\Vert T\Vert^p\left\Vert \sum\limits_{j=1}^n\varepsilon_j f_j\right\Vert_p^p\right]\right)^{1/p}\\
&\leq
A\Vert T\Vert\left(\mathbb{E}\left[\left\Vert \sum\limits_{j=1}^n\varepsilon_j f_j\right\Vert_p^p\right]\right)^{1/p}\\
&\leq
A\Vert T\Vert A \left\Vert\left(\sum\limits_{j=1}^n|f_j|^2\right)^{1/2}\right\Vert_p
\end{align}
$$
So you can put $C= A^2\Vert T\Vert$
