On the speed of convergence of iteration of a function (Q1321844)

It is shown that sometimes in the so-called indeterminate case the rate of convergence of the sequence (\(f_ n\): \(n= 1,2,\dots\)) of iterates can approximately determine the form of \(f\). More exactly, the following is proved: Let \(f: [0,c]\to \mathbb{R}\) be a continuous, increasing and concave function satisfying \(0< f(x)< x\) for \(x\in (0,c]\) (here \(c\) is a positive real). Assume that there is \(a\) such that \(\lim_{n\to\infty} n^ a f_ n(x)\) exists and is positive for every \(x\in [0,c]\). Then for every \(\varepsilon>0\) there is \(c_ \varepsilon>0\) such that \(x- x^{p- \varepsilon}\leq f(x)\leq x- x^{p+\varepsilon}\) for \(0\leq x\leq c_ \varepsilon\), where \(p= 1+ 1/a\). This is a partial converse of a classical result of A. M. Ostrowski. The assumption of concavity of \(f\) turns out to be essential for the validity of the theorem.