Minimizing this complex expression

Question

I am working through Ahlfors' Complex Analysis book. I have come to the section in Chapter 1 on inequalities. Among the exercises in this section is this:

Given complex numbers $a$ and $b$, choose $z$ such that the following expression is minimized: $$\left\lvert z-a \right\rvert^2 + \left\lvert z-b \right\rvert^2$$

It seems to me that it is relatively straightforward to think about this in terms of dot products. If one maximizes the dot product of $z$ with both $a$ and $b$, $z-a$ and $z-b$ are minimized. Therefore the squares of the norms are minimized.

The way I decided to do choose $z$ is according to the following: $$ \mathfrak{Re}(z) = \frac{\mathfrak{Re}(a)+\mathfrak{Re}(b)}{2}; \\ \mathfrak{Im}(z) = \frac{\mathfrak{Im}(a)+\mathfrak{Im}(b)}{2}; $$ Simply averaging the real and complex components of the given numbers $a$ and $b$ seems to be a relatively straightforward way of minimizing the expression. I don't think this is what Ahlfors is going for, though. This problem appears in section 1.5, complex inequalities. I am failing to draw a connection between this problem and that topic.

Answer 1

Hints: geometrically, your expression is the sum of the squares of distances from your point $z$ to $a$ and to $b$. This is a convex function with some useful symmetries.

Answer 2

Geometrically, you should expect the midpoint $(a+b)/2$ to be the minimizing point. And this suggests working $|z-(a+b)/2|^2$ into the formula somehow. So, after expanding the squares ($|z-a|^2= |z|^2 -2\operatorname{Re}(z\bar a ) + |a|^2$, etc) combine them as $$ |z-a|^2+|z-b|^2 = 2|z|^2 -2\operatorname{Re}(z\bar a + z\bar b) + |a|^2+|b|^2 $$ and recognize that this is $2|z-(a+b)/2|^2$ up to some constant term.