Subgroups with a small sum of element orders

A finite group $G$ is said to be a $\mathcal{B}_{\psi}$-group if $\psi(H)<|G|$ for any proper subgroup $H$ of $G$, where $\psi(H)$ denotes the sum of element orders of $H$. In this paper, we characterize the $\mathcal{B}_{\psi}$-groups up to the class of finite nilpotent groups. We also study the above inequality in the case of finite $P$-groups and Schmidt groups, respectively.