An Optimal Double Inequality between Seiffert and Geometric Means

Abstract

For α, β ∈ (0, 1/2) we prove that the double inequality G(αa + (1 − α)b, αb + (1 − α)a)<P(a, b) < G(βa + (1 − β)b, βb + (1 − β)a) holds for all a, b > 0 with a ≠ b if and only if and . Here, G(a, b) and P(a, b) denote the geometric and Seiffert means of two positive numbers a and b, respectively.