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The Cauchy-Schwarz inequality states $|\langle x,y \rangle |\leq ||x||\cdot ||y||.$ Use this tag for questions related to the CS inequality and its applications.

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A homogeneous but slightly asymmetric inequality

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| …