step we show that $\log |h(z)|\leq C|z|^{\rho_1} $ for all $z\in \Omega$. Recall we have assumed $\rho_1>\rho$. Here and below $C$ is a finite constant which may change from line to line. If $x\in \Real$, $1\leq |x|$, then $\log|h(x)|=\log|f(x)|\leq C_0(1+|x|^{\rho_1})$. Moreover, from estimates on canonical products, $$\label{eq:gjbound}
\log |g_j(z)| \leq C(1+|z|^\rho), \; j=1,2$$ for some constant $C$, see [@levin Lemma I.4.3].

To aid in notation, we set $\Omega_R= \{ z\in \Complex: \; {\operatorname{\rm Im}\nolimits}z \geq 1\; \text{and}\; 1\leq |z|\leq R\}$.

As is shown in the proof of [@levin Theorem I.12], given $R>0$ and $0<\delta<1$ there is an $r_j\in [R, R(1-\delta)^{-1}]$ so that for all $z\in \Complex$ with $|z|=r_j$, $$\begin{aligned}
\log |g_j(z)| & \geq -\left(2+\log \frac{12 e}{\delta}\right) \log
\max_{|z|=2 e R(1-\delta)^{-1}} |g_j(z)|,\; \text{for $j=1,\;j$}.\end{aligned}$$ Using (\[eq:gjbound\]), this gives $$\begin{aligned}
\label{eq:g2lb}
\log |g_j(z)|
& \geq -C_{ \delta} (1+(R(1-\delta)^{-1})^\rho ),\; |z|=r_j,\; j=1,\;2.\end{aligned}$$ Now fix a $\delta>0$, $\delta<1$. Given any $R>1$, we can find an $r_2\in [R,R(1-\delta)^{-1}]$ as above so that (\[eq:g2lb\]) holds for $j=2$. Then using that $$\max_{z\in \Omega_R } \log |h(z)|
\leq \max_{ z\in \Omega_{r_2}} \log |h(z)|= \max_{z\in \partial \Omega_{r_2}}
\log |h(z)|,$$ our assumptions on $f$, and (\[eq:g2lb\]) we find $$\max_{ z\in \Omega_R}\log |h(z)|
\leq C_\delta (1+ (R(1-\delta)^{-1})^{\rho_1})+ C(1+R^\rho)\leq \tilde{C}_\delta (1+R^{\rho_1}).$$

For 