Counterexamples for Hölder's inequality when $p$ and $q$ are not conjugate. Hölder's inequality shows that, when $$ \frac{1}{p} + \frac{1}{q} = 1,$$ and $f\in L^p$ and $g\in L^q$, then $$\Vert f\,g\Vert_1 \le \Vert f \Vert_p \Vert g \Vert_q.$$
Is there an example of this inequality failing for $p=q=1$?  I have been unable to find one in the usual places (analysis texts, and various combinations of search terms online).

Let $f(x)=x^2, g(x)=\frac{1}{1-x}$ on $[0,\frac{1}{2}]$
Here $\|f\|_1=\frac{1}{3}\frac{1}{8}=\frac{1}{24}$, $\|g\|_1=\ln 2$
Now $\|fg\|_1=\ln 2-\frac{5}{8}\gt\frac{\ln2}{24}$
