Sums of element orders in groups of order 2m with m odd

If G is a finite group, then psi(G) denotes the sum of orders of all elements of G and if k is a positive integer, then C_k denotes a cyclic group of order k. Moreover, psi(C_k) will be sometimes denoted by psi(k). In this paper we deal with groups of order n = 2m with m odd. Our main results are the following two theorems: Theorem 1. Let G be a non-cyclic group of order n = 2m, with m an odd integer. Then psi(G) is less or equal to 13/21 psi (Cn). Moreover, psi(G) = 13/21 psi (Cn) if and only if G = S_3xC_(n/6), where n = 6m_1 with (m_1, 6) = 1 and S_3 is the symmetric group on three letters. Theorem 2. Let Dn be the set of non-cyclic groups of the fixed order n = 2m, where m is an odd integer, and suppose that m = p_1^(a_1) p_2^(a_2)... p_t^(a_t), where p_i are distinct primes and a_i are positive integers for all i. If G is in Dn, then psi(G) is less or equal to (1/3 + 2l /3psi (l))psi (Cn), where l = min{p_i^(a_i) , i in {1,.., t}}. Moreover, G in D_n satisfies psi(G) = ( 1/3 + 2l3 (l) psi (Cn) if and only if G = D_2l x C_n/2l, where D_2l is the dihedral group of order 2l.