Inequalities for means in two variables. (English) Zbl 1020.26011

Alzer, Horst; Qiu, Song-liang

Arch. Math. 80, No. 2, 201-215 (2003).

Summary: We present various new inequalities involving the logarithmic mean \(L(x,y) = (x-y)/(\log x-\log y)\), the identric mean \(I(x,y) = (1/e)(x^x/y^y)^{1/(x-y)}\), and the classical arithmetic and geometric means, \(A(x,y) = (x+y)/2\) and \(G(x,y) = \sqrt{xy}\). In particular, we prove the following conjecture, which was published in 1986 [H. Alzer, Arch. Math. 47, 422-426 (1986; Zbl 0585.26014)]. If \(M_r(x,y) = (x^r/2+y^r/2)^{1/r}\) (\(r\neq 0\)) denotes the power mean of order \(r\), then \[ M_c(x,y) <\frac 12(L(x,y)+I(x,y)) \qquad (x,y>0, x\neq y) \] with the best possible parameter \(c=(\log 2)/(1+\log 2)\).

MSC:

26D15	Inequalities for sums, series and integrals
26E60	Means

Keywords:

arithmetic mean; geometric mean; inequalities; logarithmic mean; identric mean; power mean

Citations:

Zbl 0585.26014