Scaling Invariant Functions for Weights Let $n\in\mathbb{N}$ and $w_i\in\mathbb{R}^{+}$ for all $1\leq i\leq n$. Consider $\left(w_{1},\dots,w_{n}\right)\in\mathbb{R}^{n}$.
What are necessary and sufficient conditions for functions $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ so the following holds?
(Desired Property): For all $c\in \mathbb{R}^{+}$ there exists $\lambda \in \mathbb{R}^{+}$ such that
$$\left(f\left(cw_{1}\right),\dots,f\left(cw_{n}\right)\right)=\lambda\left(f\left(w_{1}\right),\dots,f\left(w_{n}\right)\right).$$

For context: these $w_i$ are weights, and I'm looking for functions applicable to the weights that are independent of scaling.
For example, the function $\exp:\mathbb{R}^{+}\to\mathbb{R}^{+}$ doesn't have this property, since $\left\{ e^1,e^2\right\}\neq \lambda \left\{ e^{2},e^{4}\right\} $ for all $\lambda \in \mathbb{R}^{+}$.
A sufficient condition is $f\left(xy\right)=k f\left(x\right)f\left(y\right)$ for some $k\in \mathbb{R}^{+}$ and all $x,y\in \mathbb{R}^{+}$.
Is this condition also necessary? How would one prove this / what's a counter-example? I'd appreciate any help finding the conditions, preferably explicit conditions on the functions.
 A: Note: this answer assumes that you mean, based on how you formulated your question mathematically, that for any $c \in R^+$ you can find a $λ \in R^+$ satisfying your condition, i.e. that you can have a different $λ$ for each value of $c$. If that's not what you're looking for, then the answer will change.
Taking a crack at this, I believe your functional equation is equivalent to looking for necessary and sufficient conditions for $f:R^+ \rightarrow R^+$ s.t. for every $c \in R^+$ there exists a $λ \in R^+$ s.t. $f(cx) = λf(x)$ since $f$ seems to be applied to each weight independently anyway. The only function that satisfies this is the identity function $f(x) = x$, and thus $λ = c$.
Proof:
Let an arbitrary $c \in R^+$ be given. Then we have the following by just moving the lambda to the other side:
$f(x) = f(cx)/λ$
And we can obtain this by a change of variable on the equation $f(cx) = λf(x)$:
$f(x) = λf(x/c)$
Equating the two gives us:
$f(cx)/λ = λf(x/c)$
Or:
$f(cx) = λ^2f(x/c)$
Or by another change of variable:
$f(x) = f(c^2x)/λ^2$
Which can be expanded via changes of variable to the system of functional equations:
$f(x) = f(c^{2n}x)/λ^{2n}$ for all $n \in N$
Solving for λ gives the following system of equations for all $n \in N$:
$λ = \sqrt[2n]{f(c^{2n}x)/f(x)}$
The only function $f:R^+ \rightarrow R^+$ that satisfies the above system of equations is $f(x) = x$, the identity function, and thus $λ = c$.
