## Some mean values related to the arithmetic-geometric mean.(English)Zbl 0892.26015

Let $r_n(t)= (a^n\cos^2t+ b^n\sin^2 t)^{1/n}\qquad (n\neq 0,\text{ integer});$
$r_0(t)= \lim_{n\to\infty} r_n(t)= a^{\cos^2 t}b^{\sin^2 t} \qquad (a, b>0).$ For a strictly monotonic function $$p:\mathbb{R}^+\to\mathbb{R}$$ let $$M_{p,n}(a,b)= p^{-1}\left({1\over 2\pi} \int^{2\pi}_0 p(r_n(t))dt\right)$$. For $$n\in\{-1,+1,+2\}$$ earlier investigations by H. Haruki and T. M. Rassias characterized the functions $$p$$ for which $$M_{p,n}$$ is one of the: arithmetic-geometric mean, arithmetic mean, geometric mean, or the square root mean. In this interesting paper, the author gives unique proofs for arbitrary $$n$$. For this purpose certain functional equations, recurrence relations and connections with the complete elliptic integrals are exploited.

### MSC:

 26D15 Inequalities for sums, series and integrals 33C75 Elliptic integrals as hypergeometric functions

### Keywords:

inequalities; arithmetic-geometric mean; elliptic integrals
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### References:

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