\geq \lambda^{-1}(C_gd), \text{ for } d \gg 0\},\end{aligned}$$ where $A_d$ is the subset of elements of $R[[Y]][X]$, which are homogeneous of degree $d$ in $X$, and ${\text{ord}}_Y (g_d)$ is the largest integer $k$ for which $g_d$ is an element of $Y^kR[X][[Y]]$.

It is clear that for any positive constant $c >  0$, $R[X; Y, c\lambda] = R[X; Y, \lambda]$. If $\lambda(x)$ is a positive constant, then $R[X; Y, \lambda] = R[[Y]][X]$. If $\lambda(x) = x$ for all $x$ in ${\mathbf{R}}_{\geq 0}$, then $R[X; Y, \lambda]$ is called the over-convergent subring of $R[X][[Y]]$, which is the starting point of Dwork’s $p$-adic theory for zeta functions. In both of these cases, if $R$ is noetherian, it is known that $R[X; Y, \lambda]$ is noetherian: when $\lambda$ is constant, the result follows from Hilbert’s Basis Theorem; the case in which $\lambda(x) = x$ is proved in Fulton [@Fulton]. More generally, if $R$ is noetherian and $\lambda$ satisfies the following inequality, $$\begin{aligned}
 \lambda(x) + \lambda(y) \leq \lambda(x+y) \leq \lambda(x)\mu(y)\end{aligned}$$ for all sufficiently large $x$ and $y$, where $\mu$ is another positive valued function such that $\mu(x) \geq 1$ for all $x$ in ${\mathbf{R}}_{\geq0}$, then $R[X; Y, \lambda]$ is also noetherian as shown in Wan 