Optimal bounds for the first and second Seiffert means in terms of geometric, arithmetic and contraharmonic means

摘要

In this paper, we find the greatest values α, λ and the least values β, μ such that the double inequalities α [ G ( a , b ) / 3 + 2 A ( a , b ) / 3 ] + ( 1 ? α ) G 1 / 3 ( a , b ) A 2 / 3 ( a , b ) 0 $a,b>0$ with a ≠ b $aeq b$ . Here G ( a , b ) $G(a,b)$ , A ( a , b ) $A(a,b)$ , C ( a , b ) $C(a,b)$ , P ( a , b ) $P(a,b)$ and T ( a , b ) $T(a,b)$ denote the geometric, arithmetic, contraharmonic, first Seiffert and second Seiffert means of two positive numbers a and b, respectively.