  In this paper, we consider homogeneous quasideviation means generated by real
functions (defined on $(0,\infty)$) which are concave around the point $1$ and
possess certain upper estimates near $0$ and $\infty$. It turns out that their
concave envelopes can be completely determined. Using this description, we
establish sufficient conditions for the Hardy property of the homogeneous
quasideviation mean and we also furnish an upper estimates for its Hardy
constant.
