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Is it true that for a smooth real-valued function $h(z)$ on some neighborhood of the closure of a bounded domain, that $h$ can be expressed as the difference of two smooth subharmonic functions? If so, how? If I considered $u(z)= h(z) + C|z|^2$, how would that help? I'd appreciate any input I could get. I mean clearly, $C|z|^2$ is subharmonic, isn't it?

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That's a good approach. Since $h$ is smooth on a neighborhood of the domain, $h(z)+C|z|^2$ is subharmonic if $C$ is large enough. (Note that $\Delta h$ is bounded from below and $\Delta(|z|^2) = 4$.)