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Infinitely differentiable generalized logarithmic and exponential functions. (English) Zbl 0742.39005

By a natural iterative procedure, the author constructs a C real function h satisfying the functional equation h(e x )=e h(x) -1, x. Using this h and some of his earlier results [Bull. Austr. Math. Soc. 38, No. 3, 351-356 (1988; Zbl 0643.39002) and J. Math. Anal. Appl. 155, No. 1, 93-110 (1991; Zbl 0716.39006)] he finds C solutions G of the functional equation G(e x )=G(x)+1, x.

The solution G is called a generalized logarithmic function.

MSC:
39B12Iterative and composite functional equations
33E99Other special functions
30D05Functional equations in the complex domain, iteration and composition of analytic functions