How does one determine the rate of convergence of a sequence of iterations? (Q1311129)

The paper deals with the asymptotic behavior of a sequence of iterations \(\psi^ n(x)\) where \(\psi\) is a continuous function, \(\psi: (a,b)\to (a,b)\), \(a,b\in \mathbb{R}\), \(a< b\). The asymptotic counter of \(\psi\) is defined to be a function \(F: (a,b)\to \mathbb{R}\) such that \(F\) is decreasing on \((a,\mu)\) for some \(\mu\in (a, b)\) and \(F(\psi^ n(x))\sim n\), \(n\to\infty\) for any \(x\in (a, b)\). The main result proved enables to find asymptotic counter for a class of functions which improves a result from \textit{G. Pólya} and \textit{G. Szegő} [Problems and theorems in analysis. Vol. 1 (1972; Zbl 0236.00003) (1st German ed. 1925; JFM 51.0173.01)].