function equation

hxy09

Find all functions $f$ : $R\rightarrow R$
such that $f(x^3 + y^3) = (x + y)(f^2(x) - f(x)f(y) + f^2(y))$

**Posted:** Sat Oct 24, 2009 1:21 am

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123eeee · New Member Offline Joined: 15 Jul 2009 Posts: 13

From CMO1996, we know f(rx)=rf(x)(r=p/q,p,q=1,2,3...) (1)
f(x)>=0(when x>=0) (2)
f(x^3)=xf(x)^2 (3)
Then ,for x,y>=0,2f((x^3+y^3)/2)=f(x^3+y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)>=(x+y)/2*(f(x)^2+f(y)^2)=(x+y)/2*(f(x^3)/x+f(y^3)/y)
or f(y)>=1/4(x^1/3+(2y-x)^1/3)(f(x)/x^1/3+f(2y-x)/(2y-x)^1/3),2y>=x (4)
If there are two different c's such that f(x)=c[1]x and f(y)=c[2]y ,x,y>0
from (1) and (3) ,we know there are x,y>0 such that f(x)/x>2f(y)/y,and y=x+E where E>0 and it is really small
Take it into (4) ,we will see f(2y-x)<0 ,contradiction
So f(x)=cx (x>=0,c>=0) ,then we know c=0 or 1
Similarly for x<0
So f(x)=x or f(x)=0

**Posted:** Sun Oct 25, 2009 3:00 am

hxy09

Solution

**Posted:** Fri Nov 06, 2009 6:59 am

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hxy09

**Posted:** Sat Nov 07, 2009 12:25 am

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Every thought which can solve problems should be appreciated whether it is short or long,beautiful or ugly:)