I know the generalized Hölder inequality sounds like as: Let $1\leq p_1,\ldots,p_n<\infty$ and $p>0$ such that $\frac1p=\frac1{p_1}+\cdots+\frac1{p_n}$. Then, for all measurable functions $f_1,\ldots,f_n: (X,\mu) → \mathbb C$ we have $\left\|\prod _{{k=1}}^{n}f_{k}\right\|_{p}\leq \prod _{{k=1}}^{n}\|f_{k}\|_{{p_{k}}}$. (see this or this )

My question is: what are conditions on $f_1,\ldots,f_n$ so that the equality holds, ie.. $\left\|\prod _{{k=1}}^{n}f_{k}\right\|_{p}= \prod _{{k=1}}^{n}\|f_{k}\|_{{p_{k}}}$