Sunday, September 8, 2019

Power-like functional equation




I would like to know what are the especifications of a functional equation that give us a power function as a solution.



For example, if f:RR is continuous and monotonic, such that f(x)+f(y)=f(z) iif f(λx)+f(λy)=f(λz) for all λ>0, then f(x)=axb.



Does anyone know another functional equation that gives a power function as a solution?


Answer



For all  x  and  y0,  the only continuous solutions of the equation  f(x)2=f(xy)f(x/y)  is
 f(x)=axb.



An alternative equation similar to

Cauchy's functional equation is
 f(1)f(x y)=f(x)f(y)  for all x,y.


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