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.


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