Inequalities for Chains of Normalized Symmetric Sums

In this paper we prove some inequalities between expressions of the following form: ∑ 1≤i1<···<ik≤n ai1 + · · ·+ aik a1 + · · ·+ an − (ai1 + · · ·+ aik) , where a1, · · · , an are positive numbers and k, n ∈ N, k < n. Using the results in [1] which show that ( n k ) · n−k k give a lower bound for the expressions above, we norm them and obtain the chain A(1), A(2), . . . , A(n− 1), A(n), whose terms are defined as A(k) = ∑ 1≤i1<···<ik≤n ai1+···+aik a1+···+an−(ai1+···+aik ) ( n k ) · n−k k . We prove then some inequalities between the terms of this chain. Particular cases of the results obtained in this paper represent refinements of some classical inequalities due to Nesbit[7], Peixoto [8] and to Mitrinović [5]. The results in this work are also closely related to the inequalities between complemental expressions obtained in [1].