Question

Let $f:\mathbb R^2\rightarrow\mathbb R$ be a symmetric function: $f(y,x)=f(x,y)$. It can therefore be written has a function of the elementary symmetric polynomials, here $f(x,y)=F(x+y,xy)$, where $F(\sigma,\mu)$ is defined over $4\mu\le\sigma^2$.

Let me assume that $f$ is of class $\mathcal C^r$. It is clear from the IFT that $F$ is $\mathcal C^r$ too, away from the critical line $4\mu=\sigma^2$.

What is the regularity of $F$ at the boundary ? My guess is $\mathcal C^{r/2}$ but I have been unable to prove it or to locate such a result.

For instance, if $f(x,y)=|y-x|^r$ and $r>0$ is not an integer, then $F=|\sigma^2-4\mu|^{r/2}$ is only $\mathcal C^{r/2}$ and not more.

Answer 1

Just note that $F(\sigma,\mu)=f\left(\frac{\sigma+\sqrt{\sigma^2-4\mu}}{2}, \frac{\sigma-\sqrt{\sigma^2-4\mu}}{2}\right)$.