Suppose $f$ does not vanish on $\Real \cap \Omega$, and let $n(r)$ be the number of zeros of $f$ in $\{ z:\; {\operatorname{\rm Im}\nolimits}z \geq 0, \; 1\leq |z|\leq r\}$ counted with multiplicity. Suppose that that there are constants $C_0$ and $\rho$, $\rho$ not an integer, so that $$n(r) \leq C_0(1+r^\rho )$$ and $$\left| \frac{f'(x)}{f(x)}\right| \leq C_0 (1+|x|^{\rho-1}) \; \text{ for all } 
x\in \Real \; \text{with}\; |x|\geq 1.$$ Suppose in addition that there are some constants $\rho_1$, $C_1$ so that $\log |f(z)|\leq C_1(1+|z|^{\rho_1}) $ for all $z\in \Omega$. Then there is a constant $C$ so that $|f(z)|\leq Ce^{|z|^\rho} $ for $z\in \Omega$.

We will assume $\rho_1>\rho$ as otherwise there is nothing to prove.

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

We prove this proposition by constructing a function to which we can apply Lemma \[l:p-lapp\]. Let $p$ denote the greatest integer less than $\rho$, and $\{ a_j\}$ the zeros of $f$ in $\Omega$, repeated according to multiplicity, with $|a_1|\leq |a_2|\leq ...$. Set $$h(z)=\frac{f(z)g_1(z)}{g_2(z)},$$ where $$g_1(z)= \prod_{n=1}^{\infty}  E_p(z/\overline{a_n}) \; \text{and}\; g_2(z)=
\prod_{n=1}^{\infty} E_p(z/a_n).$$ Note that $h$ is analytic in $\Omega$ and does not vanish there.

As an intermediate 