On the sum of the inverses of the element orders in finite groups

Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function m(G) = Sigma(g is an element of G) 1/o(g), where o(g) denotes the order of g is an element of G. In this paper, we prove that if m(G) < m(A(5)), m(G) < m(A(4)), m(G) < m(S-3), m(G) < m(Q(8)) or m(G) < m(C-2 x C-2), then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.