We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied earlier by the author and present two classes of special functions, namely, ultraexponential and infralogarithm f -type functions. As a result of this investigation, we obtain a general solution of the Abel equation α(f(x)) = α (x) + 1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that an infralogarithm f -type function is its unique solution. We also show that an infralogarithm f -type function is an essentially unique solution of the Abel equation. Similar theorems are proved for ultraexponential f -type functions and their functional equation β(x) = f(β(x − 1)), which can be considered as dual to the Abel equation. We also solve a certain problem unsolved before and study some properties of two considered functional equations and some relations between them.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 2, pp. 281–288, February, 2011.
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Hooshmand, M.H. Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation. Ukr Math J 63, 328–336 (2011). https://doi.org/10.1007/s11253-011-0506-z
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DOI: https://doi.org/10.1007/s11253-011-0506-z