Let $\mathcal{F}$ be the set of continuous functions $\varphi$ from $\mathbb{C}$ to $[0,1]$ that satisfy $\begin{align}\varphi(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\varphi(z+e^{i\theta})d\theta\end{align}$ for all $z\in\mathbb{C}$. For $\varphi\in\mathcal{F}$ and $\delta\gt0$, let $\varphi_\delta$ be the function defined by $\begin{align}\varphi_\delta(z)=\frac{1}{\delta^2}\int_{-\delta/2}^{\delta/2}\int_{-\delta/2}^{\delta/2}\varphi(z+\alpha+i\beta)d\alpha d\beta\end{align}$. Is the set of functions $\varphi_\delta$ compact w.r.t to the sup norm? I originally wanted to prove that $\mathcal{F}$ contains constant functions only. So far, I managed to prove that any compact subset of $\mathcal{F}$ (with some invariance under rotations and translations) contains constant functions only. If the set of $\varphi_\delta$'s is shown compact, I would achieve my original intent.
BTW. this is related to this post which has a probabilistic solution.