Exact bounds for the sum of the inverse-power of element orders in non-cyclic finite groups

Abstract: Given a finite group $G$ of order $n.$ Denote the sum of the inverse-power of element orders in $G$ by $m(G).$ Let $\mathbb{Z}_n$ be the cyclic group of order $n.$ Suppose $G$ is a non-cyclic group of order $n$ then we show that $m(G)\geq \frac{5}{4}m(\mathbb{Z}_n).$ Our result improves the inequality $m(G)>m(\mathbb{Z}_n)$ obtained by Baniasad Azad, M., and Khorsravi B. Moreover, this bound is best as for $n=4l, l$ odd, there exists a group $G$ of order $n$ satisfying $m(G)=\frac{5}{4}m(\mathbb{Z}_n)$. Moreover, we will establish that $\frac{1}{q-1}m(G)< m(\mathbb{Z}_n)\leq \frac{4}{5}m(G),$ where $G$ is a non-cyclic group of odd order.