A Sharp Double Inequality between Seiffert, Arithmetic, and Geometric Means

الملخص EN

For fixed s≥1 and any t1,t2∈(0,1/2) we prove that the double inequality Gs(t1a+(1-t1)b,t1b+(1-t1)a)A1-s(a,b)0 with a≠b if and only if t1≤(1-1-(2/π)2/s)/2 and t2≥(1-1/3s)/2.

Here, P(a,b), A(a,b) and G(a,b) denote the Seiffert, arithmetic, and geometric means of two positive numbers a and b, respectively.