Use Jensen's inequality (Example 13.14 (i), (ii)) to derive Hlder's inequality and Minkowski's inequality. Use [Lambda(x)=x^{1 /

Use Jensen's inequality (Example 13.14 (i), (ii)) to derive Hölder's inequality and Minkowski's inequality. Use

\[\Lambda(x)=x^{1 / q}, x \geqslant 0, \quad w=|f|^{p} \quad \text { and } \quad u=|g|^{q}|f|^{-p} \mathbb{1}_{\{f eq 0\}}\]

for Hölder's inequality and

\[\Lambda(x)=\left(x^{1 / p}+1ight)^{p}, x \geqslant 0, \quad w=|f|^{p} \mathbb{1}_{\{f eq 0\}} \quad \text { and } \quad u=|f|^{-p}|g|^{p} \mathbb{1}_{\{f eq 0\}}\]

for Minkowski's inequality.

Data from example 13.14 (i)

Data from example 13.14 (ii)

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(i) Taking u := wv/fw du in Theorem 13.13(1), we get for convex functions V: [0, ) [0,00) Juw dv fw du JV(u)w dv S w dv VuM(A), u>0.