Definition 5.13.2C3781B231934

Definition 5.13.2.C3781B231934 A subharmonic function on $\Omega $ is a function $\varphi \colon \Omega \rightarrow [-\infty ,\infty )$ satisfying the following three conditions:

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   $\varphi \not\equiv -\infty $;

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   $\varphi $ is upper semicontinuous;

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   $\varphi $ satisfies the sub-mean value inequality: For any $a\in \Omega $ and $r>0$ such that $\overline{B_1(a,r)}\subseteq \Omega $, we have

   \[ \varphi (a)\leq \frac{1}{2\pi }\int _0^{2\pi } \varphi (a+r\mathrm{e}^{\mathrm{i}\theta })\, \mathrm{d}\theta .{}^{\href{#sgpt-a0000001456}{1}} \]

We will denote the set of subharmonic functions on $\Omega $ by $\mathrm{SH}(\Omega )$.