Show the following improvement of Doob's maximal inequality Theorem 25.12 . Let \(\left(u_{n}ight)_{n \in \mathbb{N}}\) be a martingale or \(\left(\left|u_{n}ight|^{p}ight)_{n \in \mathbb{N}}, 1

\[\max _{n \leqslant N}\left\|u_{n}ight\|_{p} \leqslant\left\|u_{N}^{*}ight\|_{p} \leqslant \frac{p}{p-1}\left\|u_{N}ight\|_{p} \leqslant \frac{p}{p-1} \max _{1 \leqslant n \leqslant N}\left\|u_{n}ight\|_{p} .\]

Data from theorem 25.12

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Theorem 25.12 (Doob's maximal IP-inequality) Let (X, A, An, H) be a o-finite filtered measure space, 1