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

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].

### 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
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