I don't understand why we need a less than and equal sign in the last inequality.
Theorem: Let $C$ be a circle of diameter $d$ in $\Bbb{R}^2$ and let $G$ be the set of all the points inside $C$. Suppose that $u$ is a real-valued function that is defined and continuous on $G\cup C$ and is harmonic in $G$. Then the values of $u$ in $G$ cannot exceed the maximum of $u$ on $C$, nor can they be less than the minimum of $u$ on $C$.
Proof: Let $m$ be the maximum of $u$ on $C$, and suppose that at some point $(x_{0},y_{0})\in G$,$u(x_{0},y_{0})=M>m$. We may suppose that $M$ is the maximum of $u$ on $G\cup C$. Now translate the origin to $(x_{0},y_{0})$. We obtain a function we shall denote by $u$ that has all the properties of our original function and has maximum $m$ over $C$ and maximum $M$ over $G\cup C$ but attains this value at $(0,0)$. Now define an auxiliary function $v(x,y)=u(x,y)+\frac{M-m}{2d^2}(x^2+y^2)$. Clearly. $v(0,0)=M$ and, if $(x,y)\in C$, then $v(x,y)\leq m+\frac{M-m}{2d^2}d^2=\frac{M+m}{2}<M$ $\leftarrow$ this one (The proof goes on)
Shouldn't it be just less than? Or am I missing something?