1

Question -

Find all functions $f:\mathbb{R} \to [0, \infty)$ such that $$f(x^2 + y^2)=f(x^2 - y^2)+ f(2xy)$$ for all $x,y\in\mathbb{R}$.

My try -

$f(0)=0$ and $f(x)=f(-x)$ for all $x>0$ obtained by putting $y=-x$ ...

Now I replaced $x^2 +y^2$ by $x$ And $x^2 - y^2$ by $y$ .. I get $f(x)=f(y)+f(2xy)$ Putting $y=0$ I get $f(x)=0$ ...

Now there is another solution i.e $f(x)=cx^2$ but I am not able to apply any other substitution to reach at this...

Any hints ??

Batominovski
  • 49,629
IMO 2021 GOLD
  • 416

1 Answers1

5

Suppose that $f:\mathbb{R}\to\mathbb{R}_{\geq 0}$ is a function such that $$f(x^2+y^2)=f(x^2-y^2)+f(2xy)\,,\tag{*}$$ for all $x,y\in\mathbb{R}$. For each $x,y\in\mathbb{R}$, let $P(x,y)$ denote the statement (*).

From $P(0,0)$, we get $f(0)=0$. Thus, $P(0,y)$ yields $$f(+y^2)=f(-y^2)$$ for all $y\in\mathbb{R}$. Therefore, $f$ is an even function. Let $F:\mathbb{R}_{\geq 0}\to\mathbb{R}$ be the function defined by $$F(t):=f(\sqrt{t})$$ for all $t\geq 0$.

Now, since $f$ is an even function, $$\begin{align}F\big((x^2+y^2)^2\big)&=f(x^2+y^2)=f(x^2-y^2)+f(2xy)\\&=f\big(|x^2-y^2|\big)+f\big(2|xy|\big)\\&=F\big((x^2-y^2)^2\big)+F\big((2xy)^2\big)\end{align}$$ for all $x,y\in\mathbb{R}$. This shows that $$F(u+v)=F(u)+F(v)\tag{#}$$ for all $u,v\geq 0$. (Note that, for $u,v\geq 0$, there are $x,y\in\mathbb{R}$ such that $u=(x^2-y^2)^2$ and $v=(2xy)^2$.) Thus, $F$ satisfies Cauchy's functional equation and $F$ is a nonnegative function. Therefore, $F$ is a nondecreasing function. Ergo, there exists $k\geq 0$ such that $$F(t)=kt$$ for all $t\in\mathbb{R}_{\geq 0}$.

Since $f(x)=f\big(|x|\big)=F(x^2)$ for all $x\in\mathbb{R}$, we conclude that $$f(x)=kx^2$$ for all $x\in\mathbb{R}$ (where $k\geq 0$ is a constant). It is easy to see that all functions $f$ of this form obey (*).

Remark. For a function $F:\mathbb{R}_{\geq 0}\to\mathbb{R}$ such that $F$ satisfies Cauchy's functional equation (#), we can conclude that there exists a constant $k$ such that $F(t)=kt$ for all $t\geq 0$ if at least one of the following properties are known to be true:

  • $F$ is continuous at any point,
  • $F$ is continuous at one point,
  • $F$ is bounded on any bounded interval (with nonempty interior),
  • $F$ is bounded on one bounded interval (with nonempty interior),
  • $F$ is monotonic, or
  • $F$ is monotonic on one interval (with nonempty interior).

Without these properties, there are noncontinuous examples of $F$ (but such examples rely on the Axiom of Choice).

Batominovski
  • 49,629