A question of positively homogeneous functions

Question

Let $f$ be a positively homogeneous function of degree $k$, i.e., $$ f(x, \lambda y) = \lambda^k f(x, y)$$ for any $(x,y) \in \mathbb R^n \times \mathbb R^n, \lambda >0$. Then how can I show that $ \partial_x^\alpha \partial_y^\beta f(x,y)$ is positive homogeneous function of degree $k - |\beta|$? Here $\alpha, \beta$ denotes multi-indexes.

Answer 1

You want to use the chain rule. Firstly, notice that the derivatives in $x$ don't matter, so all you need to show is that $\lambda^{k - |\beta|} (\partial^{\beta}_y f)(x,y) = (\partial^{\beta}_y f) (x, \lambda y).$ Perhaps I will do the first step for you. Using the known homogeneity property,

$$ \partial^{\beta}_y f(x,y) = \frac{1}{\lambda^k} \partial_y^{\beta} [ f(x,\lambda y)].$$

Now expand the right-hand side using the chain rule.