A result on the sum of element orders of a finite group

Let G be a finite group and $$psi (G)=sum _{gin {G}}{o(g)}$$. There are some results about the relation between $$psi (G)$$ and the structure of G. For instance, it is proved that if G is a group of order n and $$psi (G)>dfrac{211}{1617}psi (C_n)$$, then G is solvable. Herzog et al. in (J Algebra 511:215–226, 2018) put forward the following conjecture: Conjecture. If G is a non-solvable group of order n, then $$begin{aligned} {psi (G)},{le },{{dfrac{211}{1617}}{psi (C_n)}}, end{aligned}$$with equality if and only if $$G cong A_5$$. In particular, this inequality holds for all non-Abelian simple groups. In this paper, we prove a modified version of Herzog’s Conjecture.