The mixed norm space $H(p,q,a)$ is the collection of functions $f$ analytic in the unit disk with finite norm $$||f||_{p,q,\alpha}=\left[\int_{0}^{1}(1-r)^{\alpha q-1}\left(\int_{0}^{2\pi}|f\left(re^{i\theta}\right)|^{p} d\theta\right)^{q/p}dr\right]^{1/q}.$$ Sufficient conditions on a family of measures $\{\mu_{r}:0< r <1\}$ on $U$ and a measure $\nu$ on $[0,1]$ are given to obtain an inequality $$||f||_{p,q,\alpha}^{q}\leq C\int{\left(\int{|f|^{p}d\mu_{r}}\right)^{q/p}\,d\nu(r)},\quad f\in H(p,q,\alpha)$$ with $C$ independent of $f$. Similar results are obtained for spaces of ``slow mean growth'' $(q=\infty)$ and the Hardy spaces ($q=\infty$, $\alpha=0$). In the case of the Bergman spaces $(p=q)$ these conditions are an improvement over those obtained in [5] and [6].