I would like to prove the sharp estimate for the $L_p$ norm to power $p$ with $1\leq p <\infty$. What is the constant $C$ here:
$$\left\|\sum_{j=1}^Jf_j\right\|^p_p\leq C\sum_{j=1}^J\|f_j\|_p^p$$
By triangle and Jensen inequalities, we have $$\left\|\frac{1}{J}\sum_{j=1}^Jf_j\right\|^p_p\leq \left(\frac{1}{J}\sum_{j=1}^J\|f_j\|_p\right)^p \leq \frac{1}{J}\sum_{j=1}^J\|f_j\|_p^p$$
Which gives $C = J^{p-1}$. Is this correct? Is this the sharpest inequality possible?