I am in the middle of the proof of the maximum principle for harmonic functions.
Given a harmonic function $u$ on the complex plane and $M_0\in \mathbb{C}$. Take $r>0$ and suppose there is an open arc $\ell$ contained in the circle $\{M_0+re^{it}\colon t\in [0,2\pi)\}$ such that $u(M) Does it follow from this that $u(M_0)\neq \frac{1}{2\pi}\int_0^{2\pi}u(M_0+re^{it})dt?$