An Optimal Inequalities Chain for Bivariate Means

Abstract. Let p ∈ R , M be a bivariate mean, and Mp be defined by Mp(a,b) = M1/p(ap,bp) (p = 0) and M0(a,b) = limp→0 Mp(a,b) . In this paper, we prove that the sharp inequalities L2(a,b) < P(a,b) < NS1/2(a,b) < He(a,b) < A2/3(a,b) < I(a,b) < Z1/3(a,b) < Y1/2(a,b) hold for all a,b > 0 with a = b , where L(a,b) = (a− b)/(loga − logb) , P(a,b) = (a− b)/[2arcsin((a−b)/(a+b))] , NS(a,b) = (a−b)/[2arcsinh ((a−b)/(a+b))] , He(a,b) = (a+ √ ab+ b)/3 , A(a,b) = (a+ b)/2 , I(a,b) = 1/e(aa/bb)1/(a−b) , Z(a,b) = aa/(a+b)bb/(a+b) and Y (a,b) = I(a,b)e1−ab/L(a,b) are respectively the logarithmic, first Seiffert, Neuman-Sándor, Heronian, arithmetic, identric, power-exponential and exponential-geometric means of a and b .