Orders of growth of real functions. (English) Zbl 1153.26001

The author defines the order \(\lambda \in\mathbb R\) of a function \(g(x)\) with respect to a function \(f(x)\) which is increasing to \(\infty \) as \(x\rightarrow \infty \) by the relation \(f(g(x))-f(x)\rightarrow \lambda \), as \(x\rightarrow \infty \). This extends the notion of order when \(\lambda = n\) as used in somewhat different language by M. Rosenlicht in [Trans. Am. Math. Soc. 299, 261–272 (1987; Zbl 0619.34057)]. Then the notions of continuity and linearity at \(\infty \) are defined and used to study order-comparability and equivalence. As an application of the obtained results, the author obtains a criterion for the uniqueness of fractional and continuous iterates of a functions. The analysis is largely based on the use of the Abel functional equation \(F(f(x)) = F(x)+1\) (used also by G. Szekeres in the study of various problems of growth, as indicated in this paper).


26A12 Rate of growth of functions, orders of infinity, slowly varying functions
39B12 Iteration theory, iterative and composite equations


Zbl 0619.34057
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