# how to prove $f^p\in L([0,1])(q \in (0,p))$ when $f^q \in L([0,1])$?

I've known that when $p>q\geq1$,then $L^p \subset L^q$,but when $q\in (0,p)$,I don't know how to prove that. When $\int_{[0,1]}|f|^pdx<\infty$,q\in (0,p),how can we get $\int_{[0,1]}|f|^qdx<\infty$ ?

Appreciate with help!

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Hölders inequality will save you . –  Theorem Nov 15 '12 at 10:22
$$\int_{[0,1]}|f|^{q}dx\leq \left(\int_{[0,1]}1^{\frac{p}{p-q}}dx\right)^{\frac{p-q}{q}}\left(\int_{[0,1]}|f‌​|^{q\frac{p}{q}}dx\right)^{\frac{q}{p}}=\left(\int_{[0,1]}|f|^{p}dx\right)^{\frac‌​{q}{p}}$$ correct? –  user39843 Nov 16 '12 at 3:55

Hint: apply Jensen's inequality with $\phi(t):=t^{\frac pq}$.