Noetherian property of subrings of power series rings II

Let R   be a commutative ring with unit. We study certain subrings R[X;Y,λ]R[X;Y,λ] of R[X][[Y]]=R[X1,…,Xn][[Y1,…,Ym]]R[X][[Y]]=R[X1,…,Xn][[Y1,…,Ym]], where λ is a nonnegative real-valued increasing function. These subrings naturally arise from studying p  -adic analytic variation of zeta functions over finite fields. In our previous work, we gave a necessary and sufficient condition for R[X;Y,λ]R[X;Y,λ] to be Noetherian when Y has more than one variable and λ grows as fast as linear. In this paper, we show that the same result holds even when Y   has only one variable. This contradicts Davis and Wan's result stating that R[X;Y,λ]R[X;Y,λ] is always Noetherian if R is a field. We however found a mistake in their proof.