Composition of Gevrey functions I am beginning to work with Gevrey functions in several variables but I have not been able to find a reference where the fact that the Gevrey classes are closed under composition is explicitly proven. 
I am interested in the dependence of the constants  involved in the definition of a Gevrey function.
More precisely, consider $g: \mathbb {R}^m \rightarrow \mathbb{R} $, $f: \mathbb {R}^d \rightarrow \mathbb{R}^m $ $C^\infty$ functions and compact sets $K_1 \subset \mathbb{R}^d$, $K_2 \subset \mathbb{R}^m $ such that $f (K_1) \subset K_2$ and
$$ \sup_{x\in K_1} |\partial ^ \alpha  f (x)| \leq CR^{|\alpha|}(\alpha!)^s $$
$$ \sup_{x\in K_2} |\partial ^ \alpha  g (x)| \leq DT^{|\alpha|}(\alpha!)^s $$
for every multi-index $\alpha \in \mathbb {N}^d$, where $C,D,R,T$ are positive constants and $s > 1$. Let $h = g \circ f$. The composition should satisfy an estimate of the form $$ \sup_{x\in K_1} |\partial ^ \alpha  h (x)| \leq EL^{|\alpha|}(\alpha!)^s $$
for every multi-index $\alpha \in \mathbb {N}^d$, and some positive constants $E,L$.
My question is, what is the relation between $E,L$ and $C,D,R,T$?
By this I am not asking for the best constants but for a simple dependence relation that will make the equality hold. I know asking for ''simple'' is not really objective but I hope you will get the idea. 
I have been trying to use the Faà di Bruno formula for the derivative of a composition and doing some estimates. I get some complicated relation for $E$ and, if I did not make any mistakes in my calculations, I can take $L$ as $T(1+DR)^d$.
As the calculations are quite involved I have been wondering if there is a simple answer for this or if there is a book or article where this question has been studied. Any ideas, references, comments are more than welcome.
Thanks in advance. 
 A: I recently ran into a special case of this problem, where $s = 1$. This answer is probably not relevant anymore for OP, but for future reference, it might be useful to show how such relations can be achieved.
Let $\Omega \subseteq \mathbb{R}^d$, $\Psi \subseteq \mathbb{R}^m$ and $\Phi \subseteq \mathbb{R}^n$ open. Suppose we have functions $f : \Omega \to \Psi$ and $g : \Psi \to \Phi$. Moreover, suppose that we have compact subsets $K_1 \subseteq \Omega$ and $K_2 \subseteq \Psi$, such that $f(K_1) \subseteq K_2$, and:
$$\forall j \in \{1, \dots, m\}, \qquad \sup_{x \in K_1, \alpha \in \mathbb{N}_0^d} \frac{|\partial^{\alpha}f_j(x)|}{R^{|\alpha|}|\alpha|!} \leq C$$
$$\forall j \in \{1, \dots, n\}, \qquad \sup_{x \in K_2, \alpha \in \mathbb{N}_0^m} \frac{|\partial^{\alpha}g_j(x)|}{T^{|\alpha|}|\alpha|!} \leq D$$
where $C,D,R,T > 0$. We find, making repetative use of the chain rule, for all $\alpha \in \mathbb{N}^d$, $x \in K_1$ and $j \in \{1, \dots, n\}$:
$$\partial^{\alpha} (g \circ f)_j(x) = \sum_{\{\pi_1, \dots, \pi_{|\pi|}\} = \pi \in \Pi_{|\alpha|}} \left[\sum_{\ell_1 = 1}^m \cdots \sum_{\ell_{|\pi|}}^m \partial^{(\ell^1, \dots, \ell_{\pi})}g_j(f(x)) \cdot \prod_{j=1}^{|\pi|} \partial^{\pi_j}f_{\ell_j}(x)\right]$$
where $\Pi_{|\alpha|}$ is the set of partitions of $\{1, \dots, |\alpha|\}$. Triangle inequality and our assumputions yield:
\begin{align*}
|\partial^{\alpha}(g \circ f)_j(x)| &\leq \sum_{(\pi_1, \dots, \pi_{|\pi|}) = \pi \in \Pi_{|\alpha|}} \left[\sum_{\ell_1 = 1}^m \cdots \sum_{\ell_{|\pi|}}^m DT^{|\pi|}|\pi|! \cdot \prod_{j=1}^{|\pi|} CR^{|\pi_j|}|\pi_j|!\right] \\
&\leq DR^{|\alpha|} \sum_{(\pi_1, \dots, \pi_{|\pi|}) = \pi \in \Pi_{|\alpha|}} (mCT)^{|\pi|}|\pi|! \cdot \prod_{j=1}^{|\pi|} |\pi_j|!
\end{align*}
From now on, we use the combinatorial argument as provided by the OP of this question. Hence, we select the partitions of $\Pi_{|\alpha|}$ with size $r$, and let $r$ run over all possible choices. Then, we let $k_1, \dots, k_r \in \mathbb{N}$ run over all all possible set sizes, and we note that we can find a partition with these set sizes in
$$\binom{|\alpha|}{k_1 \; \cdots \; k_r}$$
ways, where the above is a multinomial coefficient. As the subsets of a partition are not ordered, though, we are counting all such partitions $r!$ times, and hence we proceed with:
\begin{align*}
|\partial^{\alpha}(g \circ f)_j(x)| &\leq DR^{|\alpha|} \sum_{r=1}^{|\alpha|} (mCT)^rr! \cdot \sum_{\underset{k_1 + \cdots + k_r = |\alpha|}{(k_1, \dots, k_r) \in \mathbb{N}^r}} \frac{1}{r!}\binom{|\alpha|}{k_1 \; \cdots \; k_r} \prod_{j=1}^{r} k_j! \\
&= DR^{|\alpha|} \sum_{r=1}^{|\alpha|} (mCT)^r|\alpha|! \cdot \sum_{\underset{k_1 + \cdots + k_r = |\alpha|}{(k_1, \dots, k_r) \in \mathbb{N}^r}} 1
\end{align*}
It is now a simple combinatorial argument to show that there are $\binom{|\alpha| - 1}{r - 1}$ terms for any $r$ in the expression above, and so we finally obtain:
\begin{align*}
|\partial^{\alpha}(g \circ f)_j(x)| &\leq DR^{|\alpha|} |\alpha|! \sum_{r=1}^{|\alpha|} (mCT)^r \binom{|\alpha|-1}{r-1} \\
&= mCTDR^{|\alpha|} |\alpha|! \sum_{r=0}^{|\alpha|-1} (mCT)^r \binom{|\alpha|-1}{r} \\
&= mCTDR^{|\alpha|} |\alpha|! (1 + mCT)^{|\alpha|-1} \\
&= \frac{mCTD}{1 + mCT} \cdot \left(R(1 + mCT)\right)^{|\alpha|} \cdot |\alpha|!
\end{align*}
So, to answer OP's question, possible choices for $E$ and $L$ are:
$$E = \frac{mCTD}{1 + mCT} \qquad \mathrm{and} \qquad L = R(1 + mCT)$$
whenever $s = 1$. Probably, the above method can be extended for $s > 1$, but then I guess one needs:
$$r! \leq \frac{|\alpha|!}{k_1! \cdot \, ... \cdot k_r!}$$
whenever $k_1 + \cdots + k_r = |\alpha|$ and $k_1, \dots, k_r \geq 1$.
