  We prove that the composition of a quasi-nearly subharmonic function and a
quasiregular mappings of bounded multiplicity is quasi-nearly subharmonic.
Also, we prove that if $u\circ f$ is quasi-nearly subharmonic for all
quasi-nearly subharmonic $u$ and $f$ satisfies some additional conditions, then
$f$ is quasiconformal. Similar results are further established for the class of
regularly oscillating functions.
