An Optimal Double Inequality between Seiffert and Geometric Means - Yu-Ming Chu; Miao-Kun Wang; Zi-Kui Wang (Journal of Applied Mathematics)
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 α≤(1−1−4/π2)/2 and β≥(3−3)/6. Here, G(a,b) and P(a,b) denote the geometric and Seiffert means of two positive numbers a and b, respectively.