  Let $Y_v, v\in V,$ be $[0,1]$-valued random variables having a dependency
graph $G=(V,E)$. We show that \[ \mathbb{E}\left[\prod_{v\in V} Y_{v} \right]
\leq \prod_{v\in V} \left\{ \mathbb{E}\left[Y_v^{\frac{\chi_b}{b}}\right]
\right\}^{\frac{b}{\chi_b}}, \] where $\chi_b$ is the $b$-fold chromatic number
of $G$. This inequality may be seen as a dependency-graph analogue of a
generalised H\"older inequality, due to Helmut Finner. Additionally, we provide
applications of H\"older-type inequalities to concentration and correlation
bounds for sums of weakly dependent random variables.
