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

Let

${r}_{n}\left(t\right)={\left({a}^{n}{cos}^{2}t+{b}^{n}{sin}^{2}t\right)}^{1/n}\phantom{\rule{2.em}{0ex}}\left(n\ne 0,\phantom{\rule{4.pt}{0ex}}\text{integer}\right);$
${r}_{0}\left(t\right)=\underset{n\to \infty }{lim}{r}_{n}\left(t\right)={a}^{{cos}^{2}t}{b}^{{sin}^{2}t}\phantom{\rule{2.em}{0ex}}\left(a,b>0\right)·$

For a strictly monotonic function $p:{ℝ}^{+}\to ℝ$ let ${M}_{p,n}\left(a,b\right)={p}^{-1}\left(\frac{1}{2\pi }{\int }_{0}^{2\pi }p\left({r}_{n}\left(t\right)\right)dt\right)$. For $n\in \left\{-1,+1,+2\right\}$ 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 of real functions 33C75 Elliptic integrals as hypergeometric functions