Let $X$ be a Riemann surface. This is the definition of subharmonicity I have known (and have been trying to work with) for a while.
Here is my definition of 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 subharmonic if $P_D(u) \geq u$ 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 $Y$.)
Now, this is a bit hard to work with for what I'm trying to prove. I noticed this other definition somewhere: a function $u: Y \to \mathbb{C}$ is subharmonic if, with respect to any local coordinate $z = x+ iy$, we have $ \Delta u = \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right)\, u \geq 0. $
It's definitely not obvious that these definitions are equivalent on a Riemann surface. If the proof that these are equivalent is relatively short and if someone knows it, I'd appreciate a short explanation. If not, but if there is a reference that treats this clearly, I'd love to be directed to it. Thank you!