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Some mean values related to the arithmetic-geometric mean. (English) Zbl 0892.26015
Let $$r_n(t)= (a^n\cos^2t+ b^n\sin^2 t)^{1/n}\qquad (n\ne 0,\text{ integer});$$ $$r_0(t)= \lim_{n\to\infty} r_n(t)= a^{\cos^2 t}b^{\sin^2 t} \qquad (a, b>0).$$ 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.

26D15Inequalities for sums, series and integrals of real functions
33C75Elliptic integrals as hypergeometric functions
Full Text: DOI
[1] Aczél, J.: The notion of mean value. Norske vid. Selsk. forh. (Trondheim) 19, 83-86 (1946)
[2] Bullen, P. S.; Mitrinović, D. S.; Vasić, P. M.: Means and their inequalities. (1988) · Zbl 0687.26005
[3] Gini, C.: Means. (1958) · Zbl 0041.25811
[4] Haruki, H.: New characterizations of the arithmetic--geometric mean of Gauss and other well-known mean values. Publ. math. Debrecen 38, 323-332 (1991) · Zbl 0746.39004
[5] Haruki, H.; Rassias, T. M.: New characterizations of some mean-values. J. math. Anal. appl. 202, 333-348 (1996) · Zbl 0878.39005
[6] Vamanamurthy, M. K.; Vuorinen, M.: Inequalities for means. J. math. Anal. appl. 183, 155-166 (1994) · Zbl 0802.26009