Consider $\mathcal{C}$ the subset of functions $\varphi\colon\mathbb{C}\rightarrow[0,1]$ that satisfy the following properties:
$(1)\,\exists\,\varepsilon\colon\mathbb{R}\!\rightarrow\!(0,\infty)$ s.t. $\underset{x\rightarrow0}{\varepsilon}(x)\!\rightarrow\!0$ and $\forall\varphi\!\in\!\mathcal{C},\,\forall z_1,z_2\!\in\!\mathbb{C},\,|\varphi(z_1)\!-\!\varphi(z_2)|\leq\varepsilon(|z_1\!-\!z_2|)$.
$(2)$ The average of $\varphi$ on every unit-radius circle centered at $z$ is $\varphi(z)$.
Is it true that $\mathcal{C}$ is compact with respect to the uniform norm?
2026-04-01 00:09:40.1775002180
On the compactness of a set of functions that satisfy mean value property
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