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