# On the compactness of a set of functions that satisfy mean value property

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?

• Does $\epsilon$in (1) depends on $\phi$? – N. S. Jan 22 '18 at 22:21
• Is it one $\epsilon$ for all $\phi$ or is it for each $\phi,$ there is an $\epsilon_\phi?$ – zhw. Jan 23 '18 at 2:04
• Sorry for the lack of clarification @N.S. and @zhw! $\varepsilon$ is fixed for all members of $\mathcal{C}$ (i.e. independent of $\varphi\in\mathcal{C}$). – user2471 Jan 23 '18 at 17:03
• I found a related question, but it's unfortunately unanswered. – user2471 Jan 23 '18 at 19:00