Infimum and Supremum of a set

Question

I have to find infimum and supremum of $$a+2Re(\delta z+\sigma z^2) $$ where $\delta, \sigma \in \mathbb{C}$ and $z\in \mathbb{D}$. Here $\mathbb{D}$ is disk of radius 1.

This may be very easy question but I don't get it. If you know then please give me some idea.

Thanks in advance.

Answer 1

I suppose that $a \in \mathbb R$ and that $a, \delta$ and $ \sigma$ are fixed.

Then define $f(z)=a+\delta z +\sigma z^2$ and $u(z)=Re(f(z))=a+2Re(\delta z+\sigma z^2)$.

Since $f$ is holomorphic, $u$ is harmonic. By the min/max - principle we get

$ \inf u(\mathbb D)=\min u( \partial \mathbb D)$ and $ \sup u(\mathbb D)=\max u( \partial \mathbb D)$.

Can you procced ?