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, $\psi(G) \leq \frac{7}{11} \psi(\mathbb{Z}_n)$. 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 $\psi_k(G) \leq \frac{1+3.2^k}{1+2.4^k+2^k} \psi_k(\mathbb{Z}_n)$. 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 $G$.