boundedly homogeneous function

A function  $f:{ℝ}^n\to ℝ$,  where $n$ is a positive integer, is called boundedly homogeneous with respect to a set $\mathrm{\Lambda }$ of positive reals and a real number $r$, if the equation

$$f(\lambda \overrightarrow{x})={\lambda }^{r}f(\overrightarrow{x})$$

is true for all and $\overrightarrow{x}\in {ℝ}^n$  and  $\lambda \in \mathrm{\Lambda }$.  Then $\mathrm{\Lambda }$ is the set of homogeneity and $r$ the degree of homogeneity of $f$.

Example.  The function  $x\mapsto x^r\sin (\ln x)$  is boundedly homogeneous with respect to the set$\mathrm{\Lambda }=\{e^{2\pi \nu }\mathrm{⋮}\nu \in \mathbb{Z}\}$  and  with degree of homogeneity $r$.

Theorem.  Let  $f:{ℝ}_{+}\to ℝ$  be a boundedly homogeneous function with the degree of homogeneity $r$ and the set of homogeneity  $\mathrm{\Lambda }\supset \{1\}$.  Then $f$ is of the form

$f(x)=x^r{f}_1(\ln x)$(1)

where  $f_1:ℝ\to ℝ$  is a periodic real function depending on $f$.

Proof.  Defining  $g(x):=\frac{f(x)}{x^r}$,  we obtain

$$g(\lambda x)=\frac{f(\lambda x)}{{(\lambda x)}^r}=\frac{{\lambda }^{r}f(x)}{{\lambda }^r{x}^r}={\lambda }^{0}g(x)\mathit{\hspace{1em}}\forall \lambda \in \mathrm{\Lambda }.$$

Thus $g$ is a boundedly homogeneous function with the set of homogeneity $\mathrm{\Lambda }$ and the degree of homogeneity 0.  Moreover, define  $f_1(x):=g(e^x)$.  If  $\lambda \in \mathrm{\Lambda }\setminus \{1\}$  and  $p:=\ln \lambda$, we see that

$$f_1(x+p)=g(e^x{e}^p)=g(e^x\lambda )=g(e^x)=f_1(x)\mathit{\hspace{1em}}\forall x\in {ℝ}_{+}.$$

Therefore, $f_1$ is periodic and (1) is in .