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We do know that any finite collection of real (or complex) numbers $\left(z_{k}\right)_{k=1}^{n}$ verify:

$$\left|\sum_{k=1}^n z_k\right|\leq \sum_{k=1}^n\left|z_k\right|$$

Now given $1\leq p <\infty$ how could we generalize the above? I mean , find $C_{p}$ such that:

$$\left|\sum_{k=1}^n z_k\right|^p\leq C_p\cdot \sum_{k=1}^n\left|z_k\right|^p$$

Any help is useful!

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