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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).

MSC:

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

Citations:

Zbl 0619.34057
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