Estimating the Hardy constant of nonconcave homogenus quasideviation means

In this paper, we consider homogeneous quasideviation means generated by real functions (defined on $(0,\mathrm{\infty })$) which are concave around the point $1$ and possess certain upper estimates near $0$ and $\mathrm{\infty }$. 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.