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...

Another solution: notice that by Jensen's inequality,
$$\bigg(\frac{1}{m(E)}\int |f|^p\;dm \bigg)^{q/p} \leq \frac{1}{m(E)}\int |f|^q dm$$ which reduces to the desired claim. The probability measure we are using is $\mu(A):=m(A)/m(E)$ with our convex function $\phi(x) = x^{q/p}$ and integrand $|f|^p$.
