An exact upper bound for sums of element orders in non-cyclic finite groups

Denote the sum of element orders in a finite group $G$ by $\psi (G)$ and let $C_n$ denote the cyclic group of order $n$. Suppose that $G$ is a non-cyclic finite group of order $n$ and $q$ is the least prime divisor of $n$. We proved that $\psi (G)\le \frac{7}{11}\psi (C_n)$ and $\psi (G)<\frac{1}{q-1}\psi (C_n)$. The first result is best possible, since for each $n=4k$, $k$ odd, there exists a group $G$ of order $n$ satisfying $\psi (G)=\frac{7}{11}\psi (C_n)$ and the second result implies that if $G$ is of odd order, then $\psi (G)<\frac{1}{2}\psi (C_n)$. Our results improve the inequality $\psi (G)<\psi (C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $\psi (G)$-based sufficient conditions for the solvability of $G$.