We consider the lower $Q$-homeomorphisms with respect to $p$-modulus for $p\geqslant n$. For such classes of mappings, we establish an upper estimate of the measure of the image of balls and, as a consequence, obtain one analog of the known Ikoma–Schwartz lemma. The present estimate is a far-reaching generalization of the well-known Lavrent'ev result on the estimate of the area of the image of a disk under quasiconformal mappings. We give also the corresponding applications of these results to the Orlicz–Sobolev classes $W^{1,\varphi}_{\rm loc}$ in $\mathbb{R}^{n}$, $n\geqslant 3$, under a condition of the Calderon type on $\varphi$ and, in particular, to the Sobolev classes $W_{\rm loc}^{1,p}$ with $p>n-1$. The constructed examples of mappings demonstrate a precision of the obtained results.