Inequality of logarithm function in terms of $L^p$ norm

Question

I am struggling to show that

for $f\in L^q([0,1])$ with $q>0$,

$\int_{[0,1]}\log|f|dm\leq \log\left\| f \right\|_q $.

I even don't know how to start :( Hope I can get some help :)

Answer 1

Hint:

• $\forall p>0, \forall q\in(p,+\infty), \|f\|_p\le\|f\|_q$;
• $\lim_{q\rightarrow 0^+} \|f\|_q=\exp\left(\int_{[0,1]}\log|f(t)|\operatorname{d}t\right)$