How to prove the following inequality with different exponents?

Question

$$ \left( \sum |z_i|^a \right)^b \geq \left( \sum |z_i|^b \right)^a, \quad \forall a\in [1,2], b\in [2,+\infty) $$

Note that $a$ and $b$ are not necessarily to be integer. $z_i$ is real.

Thanks, guys.

Answer 1

If you raise both sides to $\frac{1}{ab}$ you get:

$$\left(\sum \lvert z_i\rvert^a\right)^\frac{1}{a}\ge\left(\sum \lvert z_i\rvert^b\right)^\frac{1}{b}$$

which can be written as $\lVert z\rVert_a\ge\lVert z\rVert_b$

But this is a well-known result, see for example here