Some mean values related to the arithmetic-geometric mean (Q1381586)

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.