Consider the Maass raising operator of weight $k$:

\begin{equation} R_k=iy {\partial \over \partial x}+ y {\partial \over \partial y} + \frac{k}{2}. \end{equation}

Fix now integers $k,N$ where $N \ge 1$, a Dirichlet character $ \chi (\bmod N) $ and define the space $A_{k,\chi}(N)$ as the space of smooth functions $f:H \longrightarrow \mathbb{C} $ ($H$ the upper open half-plane) which respects the following conditions:

\begin{equation} \quad f ( \alpha z)=\chi(d) \Big( \frac{cz+d}{|cz+d|} \Big)^k f(z) \quad \forall \alpha \in \Gamma_0(N) \end{equation} where $\Gamma_0(N)$ is the congruence subgroup having $c \equiv 0 \bmod N$,

Let $a \in \mathbb{Q} \cup \infty$ be a cusp and $\sigma_a \in SL_2(\mathbb{Z})$ map $\infty$ to $a$: then we impose that $ f(\sigma_a(x+iy)) $ is bounded by a power of $y$, when $y \to \infty$. This is the condition of moderate growth, as defined by Goldfeld and Hundley.

G&H want to prove that $R_k$ maps $A_{k,\chi}(N)$ to $A_{k+2,\chi}(N)$.

It is a routine computation to check that if $f$ respects condition 1 for the weight $k$, then $R_kf$ respects condition 1 for the weight $k+2$, everything else being equal.

However, how do you show that if $f$ has moderate growth, the same can be said of $R_kf$? I do not know if the esteem on the growth of $f$ gives esteems on the growth of its partial derivatives, moreover we do not have olomorphy for $f$, just smoothness.

By the way, the whole definition of moderate growth (which I copied as it is) appears a little confusing to me: I am supposing that with "bounded" we mean bounded in norm.