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

Given a finite group G of order $n.$ Denote the sum of the inverse-power of element orders in G by $m(G).$ Let Zₙ be the cyclic group of order $n.$ Suppose G is a non-cyclic group of order n then we show that m(G)≥ 5/4m(Zₙ). Our result improves the inequality m(G)>m(Zₙ) 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)=5/4m(Zₙ). Moreover, we will establish that 1/q-1m(G)< m(Zₙ)≤ 4/5m(G), where G is a non-cyclic group of odd order.