On a representation of symmetric functions in Carleman-Gevrey spaces

We study a representation $f(x)=\tilde f(\sigma_1(x),\dots, \sigma_d(x))$, $(x\in\mathbb R^d)$, of a symmetric function $f$, where $\sigma_j(x)$ is the symmetric homogeneous polynomial of degree $j$. Given a domain $\Omega$ in $\mathbb R^d$ and a non-decreasing sequence $\varphi$, the Carleman-Gevrey space $K^\varphi(\Omega)$ consists of functions $f\in C^\infty(\Omega)$ such that $|\partial_x^\alpha f(x)|\leqslant H^{|\alpha|+1}|\alpha|!\varphi(|\alpha|)$ for any bounded subdomain $\Omega'\subset\Omega$, $H_{f, \Omega'}$ being a positive constant. Let $S=\{(\sigma_1(x), \dots, \sigma_d(x)):x\in\mathbb R^d\}$.
Theorem. Let $\varphi$ and $\psi$ be non-decreasing sequences. Then for every symmetric $f\in K^\varphi(\mathbb R^d)$ there is $\tilde f\in K^\psi(S)$ if and only if $\psi(n)\geqslant\varphi(nd)\varepsilon^{n+1}$, $\varepsilon$ being a positive number not depending on $n$.