The solvability of a finite group by the sum of powers of element orders

We prove a new criterion for the solvability of the finite groups, depending on the function ${\psi }_k(G)$ which is defined as the sum of $k$-th powers of the element orders of $G$. We show that our result can be used to show the solvability of some groups for which the solvability does not follow from earlier similar kind of results and we emphasize the following: looking at ${\psi }_k(G)$ for $k>1$ can be useful to get further pieces of information about the group $G$.