all $z\in \Omega$.

Consider the function $h(z)= g(z) / (i+z)^{\rho}$. This is an analytic function in a neighborhood of $\Omega$ and is bounded on $\partial \Omega$. Thus, by a version of the Phragmén-Lindellöf theorem (proved, for example, by an easy modification of the proof of [@conway Corollary VI.4.2] using [@conway Theorem VI.4.1]), $h$ is bounded in $\Omega$. This implies that for $z\in \Omega$, $|f(z)| = \exp ({\operatorname{\rm Re}\nolimits}g(z)) 
\leq \exp (C(1+|z|)^{\rho})$ for some constant $C$.

We shall use the notation $E_0(z) =1-z$ and $E_p(z) = (1-z)\exp(z+z^2/2+...+ z^p/p)$ for $p\in \Natural$ for a canonical factor. The proof of the following lemma bears many similarities to proofs for estimates of canonical products; see for example [@levin Lemma I.4.3].

\[l:cproductreals\] Let $\{a_j\}\subset \Complex$ be a set of not necessarily distinct points in the open upper half plane, with $|a_1|\leq |a_2|\leq ...$ and suppose for some constants $C_0$ and $\rho$ $$n(r){\stackrel{\rm{def}}{=}}\# \{ j :\; |a_j|\leq r \}
 \leq C_0 r^\rho\; \text{ when} \; r\geq 1.$$ Suppose $\rho > 0$ is not an integer, let $p$ be the greatest integer less than $\rho$, and set $$f(z)=\prod_{n=1}^{\infty} \frac{E_p(z/\overline{a_n})}{ E_p(z/a_n)}.$$ Then for $x\in \Real$ $$\left|\int_0^x \frac{f'(t)}{f(t)} dt\right| = O(|x|^\rho)$$ as $|x|\rightarrow \infty$.

We note first that our 