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Zbl 0892.26015
Some mean values related to the arithmetic-geometric mean.
(English)
[J] J. Math. Anal. Appl. 218, No.2, 358-368 (1998). ISSN 0022-247X

Let $$r_n(t)= (a^n\cos^2t+ b^n\sin^2 t)^{1/n}\qquad (n\ne 0,\text{ integer});$$ $$r_0(t)= \lim_{n\to\infty} r_n(t)= a^{\cos^2 t}b^{\sin^2 t} \qquad (a, b>0).$$ For a strictly monotonic function $p:\bbfR^+\to\bbfR$ 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.
[József Sándor (Cluj-Napoca)]
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions
33C75 Elliptic integrals as hypergeometric functions

Keywords: inequalities; arithmetic-geometric mean; elliptic integrals

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