Proof that $\frac{3-x^{4}}{8x^{2}}$ is the only solution to the functional equation $f(x) + 3f\left(\frac{1}{x}\right) = x^{2}$

Question

How can we prove that $f(x) + 3f\left(\frac{1}{x}\right) = x^{2}$ has only one solution and no other solution and that solution is $\frac{3-x^{4}}{8x^{2}}$.

I know that this is a linear function equation, but can't find any sources which say that all linear functional equation will have only one solution.

Do you need to find out how to arrive at this solution or is it a given and you just need to find that it is unique? — Derek Luna, Nov 04 '20 at 11:33

score 2 · Accepted Answer · answered Nov 04 '20 at 11:45

2

If in addition $g(x) + 3g(1/x) = x^2$ then $$f(x) - g(x) + 3f(1/x) - 3g(1/x) = 0.$$ Define $h = f-g$ so that $$h(x) = -3 h(1/x).$$ Now swap $x \mapsto 1/x$ to find $$h(1/x) = -3h(x)$$ so that $$h(x) = 9 h(x).$$ This forces $h = 0$ and thus $f=g$.

answered Nov 04 '20 at 11:45

Umberto P.

52,165

Thanks :), what about the general case of all linear functional equations, do they all have at most one solution? – Abhishek Kaushik Nov 04 '20 at 11:49
Techniques I know but couldn't piece together...well done. – Derek Luna Nov 04 '20 at 11:49
@AbhishekKaushik do you suppose that the functional equation $f(x) = f(x+2\pi)$ has at most one solution? – Umberto P. Nov 04 '20 at 15:18
No, as all trigonometric functions will follow that. :p Thanks, I was doing equations like above repeatedly so my perception got skewed – Abhishek Kaushik Nov 04 '20 at 17:53

score 1 · Answer 2 · answered Nov 04 '20 at 11:37

1

We always have $x \ne 0.$

From

$$(1) \quad f(x) + 3f(\frac{1}{x}) = x^{2}$$

we get

$$(2) \quad f( \frac{1}{x})+3f(x)=\frac{1}{x^2}.$$

If we add $(1)$ and $(2)$ we derive

$$(3) \quad f(x)+f(\frac{1}{x})= \frac{1}{4}(x^2+\frac{1}{x^2}).$$

$(3) - (1)$ shows that

$$(4) \quad -2f(\frac{1}{x})=\frac{1}{4}(x^2+\frac{1}{x^2})-x^2.$$

Now substitute $\frac{1}{x} \to x$ in $(4)$. Easy computations give now the desired result.

answered Nov 04 '20 at 11:37

Fred

77,394

But this doesn't prove no other solution exist, or does it? – Abhishek Kaushik Nov 04 '20 at 11:38
It proves that there is no other solution ! I have shown: if $f(x) + 3f(\frac{1}{x}) = x^{2}$, then $f(x)=\frac{3-x^{4}}{8x^{2}}.$ – Fred Nov 04 '20 at 11:45
Multiplying (2) with 3 and subtracting (1) would give the result a bit more directly. – Martin R Nov 04 '20 at 11:50

Proof that $\frac{3-x^{4}}{8x^{2}}$ is the only solution to the functional equation $f(x) + 3f\left(\frac{1}{x}\right) = x^{2}$

2 Answers2