Find all harmonic functions $\phi$ in the unit disk $D= \{\ z \in \mathbb{C} : |z|<1 \}\ $ that satisfy $\phi(\frac{1}{2})=4$ and $\phi(z)\ge 4$ for all $z \in D$.
Through $\phi$ being harmonic, we get a lot of information. Writing out $\phi$ as
$$\phi(x+iy)=u(x,y)+iv(x,y)$$ where $u,v$ are real valued functions lets us apply both the definion of a harmonic function, satisfying Laplace's Equation, $$\frac{d^2\phi}{dx^2} + \frac{d^2\phi}{dy^2}=0$$ as well as the Cauchy-Riemann equations, $$\frac{du}{dx}=\frac{dv}{dy}, \frac{du}{dy}=-\frac{dv}{dx}$$
But I'm unsure on how to apply the conditions given to these equations. I have the simplest solution, $\phi(z)=4$. How can I use the two conditions, $\phi(\frac{1}{2})=4$ and $\phi(z)\ge4, \forall z\in D$ to see other solutions, or determine there are no other solutions?
Apply the maximum principle to $-\phi$, or equivalently, the minimum principle to $\phi$, and use that $\frac{1}{2}$ is in the interior of $D$.