Geodesic
A Characterization of Exponential Functions with Non-Linear Exponents
Canadian mathematical bulletin, Tome 18 (1975) no. 2, pp. 277-281

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It is well known (see e.g. [1]) that the Cauchy functional equation characterizes the function f:x→exx.It was mentioned in [2D that the functions f:x→A exp(αx2m), g:x→exx/A can be characterized by the equation (1) but the proof was done only for m=l which was considerably simple. 
Światak, Halina. A Characterization of Exponential Functions with Non-Linear Exponents. Canadian mathematical bulletin, Tome 18 (1975) no. 2, pp. 277-281. doi: 10.4153/CMB-1975-052-5
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[1] 1. Aczél, J., Lectures on functional equations and their applications, Academic Press, New York and London, 1966. Google Scholar

[2] 2. Światak, H., On a class of functional equations with several unknown functions, Aequationes Math, (to appear). Google Scholar

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