Decomposition of analytic functions on the upper half plane

Question

Let $$h:\mathbb{H}:=\{z\in \mathbb{C}:\text{Im}(z)>0\}\mapsto \mathbb{H}\cup \mathbb{R}$$ be an analytic function. We decompose $h$, using partial fraction decomposition, into two parts $h_1$ and $h_2$, one with poles in the lower half plane ($h_1$) and one with poles on the real line ($h_2$). The question is whether both parts satisfy $$\text{Im}(h_1(z))\geq0$$ and $$\text{Im}(h_2(z))\geq 0,$$ for $z\in \mathbb{H}.$ I know that $h_2$ does indeed satisfy this property. This is true because poles on the real line are simple and the coefficient appearing in front of them is non-positive, hence they are of the form $$\frac{-p}{z-t_0}$$ for $p\geq 0$ and $t_0\in \mathbb{R}$ which satisfies the property. I am stuck showing the result for $h_1$. One attempt has been to compute directly to find relation between the imaginary part of $h_1$ and $h_2$, but this has not gotten me far, it may be that $h_1$ has really tiny imaginary part and $h_2$ has a bigger one. A different attempt was to consider the argument instead. This is the case since if we consider the logarithm function with the principal cut then $\text{Im}\log(h(z))=\arg(h(z))\geq 0$ for $z\in\mathbb{H}$. The idea here is that I could get a contradiction if the argument of $h_2$ is less than 0 at some point $z\in \mathbb{H}$. Still, I have not managed to make any progress.