Constant functions

Let f , g , h be three functions from the set of positive real numbers to itself satisfying f(x)g(y)=h((x2+y2)12) for all positive real numbers x , y . Show that f(x)g(x) , g(x)h(x) and h(x)f(x) are all constant functions .

I have proved that f(x)g(x) is constant and can see that proving either of the last two will prove the final one , but I am not able to prove any of the last two .

Thanks for any help .

Answer

forall x,y,z>0, h(x2+y2)f(z)=f(x)g(y)f(z)=f(x)h(y2+z2),
thus h(x2+y2)h(z2+y2)=f(x)f(z).

Therefore, forall x,y,z,t>0 : h(x2+z2)h(y2+z2)=f(x)f(y)=h(x2+z2+t2)h(y2+z2+t2)=f(x2+z2)f(y2+z2), thus h(x2+z2)f(x2+z2)=h(y2+z2)f(y2+z2), which proves that h/f is a constant function.

Attribution
Source : Link , Question Author : Ester , Answer Author : mercio

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