how to show $\|f\|_p\leq (m(E))^{1/p-1/q}\|f\|_q$ 
If $m(E)<\infty$ and $f\in L_q(E)$, show that $\|f\|_p\leq (m(E))^{1/p-1/q} \|f\|_q$, for $1\leq p< q<\infty$ (and consequently $L_q(E)\subset L_p(E)$, whenever $m(E)<\infty$)

How to start such a problem?
It seems clear that some sort of version of Holder's inequality will have to be employed...

Hint: let $r = q/p$ and $r'$ be the conjugate exponent of $r$. Note that $|f|^p \in L_r(E)$ and $\chi_E \in L_{r'}(E)$.
