Remarks about the square equation : functional square root of a logarithm - Maciuk, Arkadiusz; Smoluk, Antoni (Wydawnictwo Uniwersytetu Ekonomicznego we Wrocławiu, 2016)
The study shows that the functional equation f (f (x)) = ln(1 + x) has a unique result in a semigroup of power series with the intercept equal to 0 and the function composition as an operation. This function is continuous, as per the work of Paulsen [2016]. This solution introduces into statistics the law of the one-and-a-half logarithm. Sometimes the natural growth processes do not yield to the law of the logarithm, and a double logarithm weakens the growth too much. The law of the one-and-a-half logarithm proposed in this study might be the solution
Mathematical Economics, 2016, Nr 12 (19), s. 39-52
Keywords: functional square root; fractional iteration; power series; logarithm; semigroup