91Ó°ÊÓ

Homogeneous functions and Euler's identity. A function \(f: G \rightarrow \mathbb{R}\) defined in some domain \(G \subset \mathbb{R}^{m}\) is called homogeneous (resp. positive-homogeneous) of degree \(n\) if the equality $$ f(\lambda x)=\lambda^{n} f(x) \quad\left(\text { resp. } f(\lambda x)=|\lambda|^{n} f(x)\right) $$ holds for any \(x \in \mathbb{R}^{m}\) and \(\lambda \in \mathbb{R}\) such that \(x \in G\) and \(\lambda x \in G\) A function is locally homogeneous of degree \(n\) in the domain \(G\) if it is a homogeneous function of degree \(n\) in some neighborhood of each point of \(G\). a) Prove that in a convex domain every locally homogeneous function is homogeneous. b) Let \(G\) be the plane \(\mathbb{R}^{2}\) with the ray \(L=\left\\{(x, y) \in \mathbb{R}^{2} \mid x=2 \wedge y \geq 0\right\\}\) removed. Verify that the function $$ f(x, y)= \begin{cases}y^{4} / x, & \text { if } x>2 \wedge y>0 \\ y^{3}, & \text { at other points of the domain, }\end{cases} $$ is locally homogeneous in \(G\), but is not a homogeneous function in that domain. c) Determine the degree of homogeneity or positive homogeneity of the following functions with their natural domains of definition: $$ \begin{aligned} f_{1}\left(x^{1}, \ldots, x^{m}\right) &=x^{1} x^{2}+x^{2} x^{3}+\cdots+x^{m-1} x^{m} \\ f_{2}\left(x^{1}, x^{2}, x^{3}, x^{4}\right) &=\frac{x^{1} x^{2}+x^{3} x^{4}}{x^{1} x^{2} x^{3}+x^{2} x^{3} x^{4}} \\ f_{3}\left(x^{1}, \ldots, x^{m}\right) &=\left|x^{1} \cdots x^{m}\right|^{l} \end{aligned} $$ d) By differentiating the equality \(f(t x)=t^{n} f(x)\) with respect to \(t\), show that if a differentiable function \(f: G \rightarrow \mathbb{R}\) is locally homogeneous of degree \(n\) in a domain \(G \subset \mathbb{R}^{m}\), it satisfies the following Euler identity for homogeneous functions: $$ x^{1} \frac{\partial f}{\partial x^{1}}\left(x^{1}, \ldots, x^{m t}\right)+\cdots+x^{m} \frac{\partial f}{\partial x^{m}}\left(x^{1}, \ldots, x^{m}\right) \equiv n f\left(x^{1}, \ldots, x^{m}\right) $$ e) Show that if Euler's identity holds for a differentiable function \(f: G \rightarrow \mathbb{R}\) in a domain \(G\), then that function is locally homogeneous of degree \(n\) in \(G\). Hint: Verify that the function \(\varphi(t)=t^{-n} f(t x)\) is defined for every \(x \in G\) and is constant in some neighborhood of \(1 .\)