Prove Equality in Holder's Inequality Given these Restrictions Consider Holder's Inequality, which states that given a measurable set $E$, $1\leq p<\infty$, and $q$ the conjugate of $p$ (defined by $\frac{1}{p}+\frac{1}{q}=1$), if $f\in L^p(E)$ and $g\in L^q(E)$, then $f\cdot g\in L^1(E)$ and $||fg||_1\leq ||f||_p||g||_q$.
We want to see that $||fg||_1 = ||f||_p||g||_q$ if and only if there exist constants $\alpha$ and $\beta$ not both zero such that $\alpha|f|^p=\beta|g|^q$ a.e. on $E$.
I really don't know where to start with this proof and would appreciate help!

Assume that $\|f\|_p=\|g\|_q=1$. Then you have $\|fg\|_1=1$. That is, using Young's Inequality,
$$
1=\int_E |fg|\leq \,\int_E\left(\frac{|f|^p}p+\frac{|g|^q}q\right)=\frac{\|f\|_p^p}p+\frac{\|g\|_q^q}q=1. 
$$
Thus 
$$
\int_E\left(\frac{|f|^p}p+\frac{|g|^q}q-|fg|\right)=0.
$$
But the integral is non-negative (due to Young's Inequality) and so 
$$
|fg|=\frac{|f|^p}p+\frac{|g|^q}q
$$
almost everywhere. Now you are reduced to the case of equality in Young's Inequality. 
