Let G be a finite group and
$\psi (G) = \sum _{g \in G} o(g)$
, where
$o(g)$
denotes the order of
$g \in G$
. There are many results on the influence of this function on the structure of a finite group G.
In this paper, as the main result, we answer a conjecture of Tărnăuceanu. In fact, we prove that if G is a group of order n and
$\psi (G)>31\psi (C_n)/77$
, where
$C_n$
is the cyclic group of order n, then G is supersolvable. Also, we prove that if G is not a supersolvable group of order n and
$\psi (G) = 31\psi (C_n)/77$
, then
$G\cong A_4 \times C_m$
, where
$(m, 6)=1$
.
Finally, Herzog et al. in (2018, J. Algebra, 511, 215–226) posed the following conjecture: If
$H\leq G$
, then
$\psi (G) \unicode[stix]{x02A7D} \psi (H) |G:H|^2$
. By an example, we show that this conjecture is not satisfied in general.