I need to prove or disprove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$ \begin{equation} \left|\left|\sum_{j=1}^l z_j\right|^p-\sum_{j=1}^l\left|z_j\right|^p\right|\leq C_{p,l}\sum_{i\neq j}\left|z_i\right|\left|z_j\right|^{P-1}. \end{equation}

I need to prove the following inequality: for any $Z=(z_1,\dots,z_l)\in\mathbb{C}^l$ for any $p\geq 2$ and $l\geq 2$ \begin{equation} \left|\left|\sum_{j=1}^l z_j\right|^p-\sum_{j=1}^l\left|z_j\right|^p\right|\leq C_{p,l}\sum_{i\neq j}\left|z_i\right|\left|z_j\right|^{P-1}. \end{equation}