Let f , g , h be three functions from the set of

positive real numbersto 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*