Let $g_+:\mathbb R \rightarrow \mathbb C $ be a function such that $\int_{-\infty}^{\infty}|g(t)|dt$ is finite, $g_+$ is not the zero function nor the the function zero except in a zero measure set, and $\forall t <0$ $g_+(t)=0$. Let $\lambda\in \mathbb R$ and $f= \lambda + \int_0^{\infty}g_+(t)e^{iyt}dt$. Suppose $f$ is an entire function and that there are positive real numbers, $a$ and $A$, such that for all $z\in \{x+iy \in \mathbb C \mid y \leq 0 \}$ $|f(z)| \leq A e^{a|Im(z)|}$.
I found the following claim for $\lambda = 1$. Since the function $f$ converges uniformly to 1 as $z \to \infty$ in $\{x+iy \in \mathbb C \mid y \geq 0 \}$ and is entire, then it can only have a finite number of zeros in $\{x+iy \in \mathbb C \mid y \geq 0 \}$.
My question is how to prove that claim. I think I can prove it for $\{x+iy \in \mathbb C \mid y > 0 \}$ (I want it for the closure) using the following steps ( I don't mind another road, this is the closest one I have found, or any help or bibliography would be very much appreciated).
1.-Prove for all $\theta \in [0,\pi]$ $h_f(\theta) = \limsup_{r \to \infty} \frac{ln|f(re^{i\theta})|}{r} = 0$, where $h_f$ is the indicator function.
2.- Use the following theorem, to obtain that: $\lim_{r \to \infty} \frac{n(r,0,\pi)}{r} = 0$, where $n(r,0,\pi)$ is the number of zeros from $\{x+iy \in \mathbb C \mid y \geq 0 \}$, the function $f$ is of exponential type and so its order is one.
The theorem I previously talked about is. If a holomorphic function $F(z)$ of order $\rho(r)$ has completely regular growth within an angle $(\theta_1, \theta_2)$, then for all values of $\alpha_1$ and $\alpha_2$ $( \theta_1 < \alpha_1 < \alpha 2 < \theta_2)$ except possibly for a denumerable set the following limit exists: $$ \frac{1}{2\pi \rho}s_F(\alpha_1,\alpha_2) = \lim_{r \to \infty} \frac{n(\alpha_1,\alpha_2)}{r^{\rho(r)}}, $$ where $$s_F(\alpha_1,\alpha_2) = \left[ h_F'(\alpha_2)-h_F'(\alpha_1)+\rho^2 \int_{\alpha_1}^{\alpha_2}h_F(\phi)d\phi \right]. $$ The exceptional denumerable set can only consist of points for which $h_F'(\alpha+0) \neq h_F'(\alpha-0)$.
Thank you very much for taking your time to read my question.