A lower bound for ratio of power means

holds for r > 0 and n∈N. We call the left-hand side of this inequality Alzer’s inequality [1] and the right-hand side Martins’ inequality [8]. Let {ai}i∈N be a positive sequence. If ai+1ai−1 ≥ ai for i ≥ 2, we call {ai}i∈N a logarithmically convex sequence; if ai+1ai−1 ≤ ai for i≥ 2, we call {ai}i∈N a logarithmically concave sequence. In [2], Martins’ inequality was generalized as follows: let {ai}i∈N be an increasing, logarithmically concave, positive, and nonconstant sequence satisfying (a +1/a ) ≥ (a /a −1) −1 for any positive integer > 1, then ( (1/n) ∑n i=1a r i ( 1/(n+m))∑n+m i=1 ai )1/r < n √ an! n+m√an+m! , (1.2)