# If a function $f$ is sufficiently close to a strictly subharmonic function $g$, then $f$ is also strictly subharmonic?

In my setting I'm working on an open subset $$Y$$ of a Riemann surface, but I don't think that matters too much. I have a smooth function $$f: Y \to \mathbb{R}$$ that is strictly subharmonic. If another smooth function $$g: Y \to \mathbb{R}$$ is sufficiently close (in the Whitney $$C^\infty$$ topology), then will it also be strictly subharmonic? I'm finding this very difficult to show with my definition of strictly subharmonic (if this result is true at all...which I suspect it is, because this topology is that strong).

Here is my definition of strictly subharmonic: Let $$A$$ denote the set of all relatively compact, open subsets of $$Y$$ on which a solution to the Dirichlet problem exists for any arbitrary continuous boundary condition. If $$D \in A$$ and $$u: Y \to \mathbb{R}$$ is continuous, denote by $$P_D(u): \overline{D} \to \mathbb{R}$$ the function which solves the Dirichlet problem on $$\overline{D}$$ (with boundary values $$u|_{\partial D}$$), and coincides with $$u$$ on $$Y \setminus D$$. A continuous function $$u: Y \to \mathbb{R}$$ is strictly subharmonic if $$P_D(u) > u$$ on $$Y$$ for all $$D \in A$$.

(By the Dirichlet problem I mean the following. Let $$q: \partial D \to \mathbb{R}$$ be a continuous function. We wish to find a continuous function $$u: \overline{D} \to \mathbb{R}$$ which coincides with $$q$$ on $$\partial Y$$ and is harmonic in $$Y$$.)