  Denote by $G$ a finite group and by $\psi(G)$ the sum of element orders in
$G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order
$t$ and write $\psi(t)=\psi(C_t)$. In this paper we proved the following
Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the
smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements
hold. (1) If $q=3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {85}{301}$, and
equality holds if and only if $n=3\cdot 7\cdot m_1$ with $(m_1,42)=1$ and
$G=(C_7\rtimes C_3)\times C_{m_1}$, with $C_7\rtimes C_3$ non-abelian. (2) If
$q>3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {p^4+p^3-p^2+1}{p^5+1}$,
where $p$ is the smallest prime bigger than $q$ and equality holds if and only
if $n=qp^2m_1$ with $(m_1,p!)=1$ and $G=C_q\times C_p\times C_p \times
C_{m_1}$.
