On certain inequalities for means. III. (English) Zbl 0976.26015

Sándor, J.

Arch. Math. 76, No. 1, 34-40 (2001).

A typical result offered is \[ A^{2/3}G^{1/3}<P<\frac{2A+G}{3}, \] where \[ P(x,y)=\frac{x-y} {4\arctan (x^{1/2}y^{-1/2})-\pi}, \] introduced by H.-J. Seiffert [e.g., Nieuw Arch. Wisk. (4) 13, No. 2, 195-198 (1995; Zbl 0830.26008)], and \(A(x,y)\) and \(G(x,y)\) are the arithmetic and geometric means, respectively, for positive reals \(x\neq y\).
[For Part I and II see J. Sándor, J. Math. Anal. Appl. 189, No. 2, 602-606 (1995; Zbl 0822.26014) and ibid. 199, No. 2, 629-635 (1996; Zbl 0854.26013), respectively].

Reviewer: János Aczél (Waterloo/Ontario)

MSC:

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

Keywords:

inequalities; arithmetic mean; geometric mean

Citations:

Zbl 0822.26014; Zbl 0830.26008; Zbl 0854.26013