Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation

Abstract

We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied by the author earlier and present two classes of special functions, namely, ultraexponential and infralogarithm $f$-type functions. As a result of present investigation, we obtain general solution of the Abel equation $\alpha (f(x)) = \alpha (x) + 1$ under some conditions on a real function $f$ and prove a new completely different uniqueness theorem for the Abel equation stating that the infralogarithm $f$-type function is its unique solution. We also show that the infralogarithm $f$-type function is an essentially unique solution of the Abel equation. Similar theorems are proved for the ultraexponential $f$-type functions and their functional equation $\beta(x) = f(\beta(x − 1))$ which can be considered as dual to the Abel equation. We also solve certain problem being unsolved before, study some properties of two considered functional equations and some relations between them.