A Criterion for Solvability of a Finite Group by 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 $g\in G$. In [M. Herzog, et. al., Two new criteria for solvability of finite groups, J. Algebra, 2018], the authors put forward the following conjecture: $\mathbf{\text{Conjecture.}}$ \textit{If $G$ is a group of order $n$ and $\psi (G)>211\psi (C_n)\mathrm{/}1617$, where $C_n$ is the cyclic group of order $n$, then $G$ is solvable.} In this paper we prove the validity of this conjecture.