An exact upper bound for the sum of powers of element orders in non-cyclic finite groups

For a finite group $G$, let $\psi (G)$ denote the sum of element orders of $G$. This function was introduced by Amiri, Amiri, and Isaacs in 2009 and they proved that for any finite group $G$ of order $n$, $\psi (G)$ is maximum if and only if $G\simeq {\mathbb{Z}}_n$ where ${\mathbb{Z}}_n$ denotes the cyclic group of order $n$. Furthermore, Herzog, Longobardi, and Maj in 2018 proved that if $G$ is non-cyclic, $\psi (G)\le \frac{7}{11}\psi ({\mathbb{Z}}_n)$. Amiri and Amiri in 2014 introduced the function ${\psi }_k(G)$ which is defined as the sum of the $k$-th powers of element orders of $G$ and they showed that for every positive integer $k$, ${\psi }_k(G)$ is also maximum if and only if $G$ is cyclic. In this paper, we have been able to prove that if $G$ is a non-cyclic group of order $n$, then ${\psi }_k(G)\le \frac{1+{3.2}^k}{1+{2.4}^k+2^k}{\psi }_k({\mathbb{Z}}_n)$. Setting $k=1$ in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for ${\psi }_k(G)$ is also obtained for finite abelian $p$-groups $G$, using which one can explicitly find out the exact value of ${\psi }_k(G)$ for finite abelian groups $G$.