On Two Conjectures about the Sum of Element Orders

Let $G$ be a finite group and $\psi (G)={\sum }_{g\in G}o(g)$, where $o(g)$ denotes the order of $g\in G$. First, we prove that if $G$ is a group of order $n$ and $\psi (G)>31\psi (C_n)\mathrm{/}77$, where $C_n$ is the cyclic group of order $n$, then $G$ is supersolvable. This proves a conjecture of M.~{T\u{a}rn\u{a}uceanu}. Moreover, M. Herzog, P. Longobardi and M. Maj put forward the following conjecture: If $H\le G$, then $\psi(G) \leqslant \psi(H) |G:H|^2$. In the sequel, by an example we show that this conjecture is not satisfied in general.