Here's the problem statement:

I have no problem showing that $\displaystyle f\in L_q$. It can also be shown that $\displaystyle f\in L_{\infty}$, in case that helps. However, I cannot show that the inequalityIf $\displaystyle f\in L_p$ for some $\displaystyle 0<p<\infty$, and every set of positive measure in $\displaystyle X$ has measure at least $\displaystyle m$, show that $\displaystyle f\in L_q$ for all $\displaystyle p<q\leq\infty$, with $\displaystyle \lVert f\rVert_q\leq m^{\frac{1}{q}-\frac{1}{p}}\lVert f\rVert_p$.

$\displaystyle \lVert f\rVert_q\leq m^{\frac{1}{q}-\frac{1}{p}}\lVert f\rVert_p$

holds. Any help would therefore be much appreciated !

(This is exercise 1.3.12 from this book, p41.)