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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-cyclicfinite group of order $n$ and $q$ is the least prime divisor of $n$. We provedthat $\psi(G)\leq\frac 7{11}\psi(C_n)$ and $\psi(G)<\frac 1{q-1}\psi(C_n)$. Thefirst result is best possible, since for each $n=4k$, $k$ odd, there exists agroup $G$ of order $n$ satisfying $\psi(G)=\frac 7{11}\psi(C_n)$ and the secondresult implies that if $G$ is of odd order, then $\psi(G)<\frac 12\psi(C_n)$.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 obtainedby S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wuin 2015. Furthermore, we obtained some $\psi(G)$-based sufficient conditionsfor the solvability of $G$.

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