Fractional iteration of exponentially growing functions

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Abstract
The fractional iteration ofexand solutions of the functional equation have frequently been discussed in literature. G. H. Hardy has shown (in [3], and in greater detail in [4]) that the asymptotic behaviour of the solutions of (1) cannot be expressed in terms of the logarithmico-exponential scale, although they are comparable with each member of the scale.1Hence solutions of (1) provide a remarkably simple instance of functions whose manner of growth does not fit into the scale ofL-functions but requires non-elementary orders of infinity for an accurate representation. This raises quite naturally the question whether there exists a most regularly growing solution of equation (1) which might serve as a prototype for this kind of growth. In a slightly more general context we may ask whether there exists a ‘best’ family of fractional iteratesfσ(x), satisfying .

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