Maximum principle application

Question

Assume $f(z)=\dfrac{z^2}{z+2}.$ I want to find: $$\begin{equation} \max_{|z|\leq1}\bigg|\frac{z^2}{z+2}\bigg| \end{equation}$$

Taking the maximum principle into consideration, it must be true that: $$|z|=1\iff z=e^{i\phi}.$$ Thus: $$ \big|f(z)\big|=\bigg|\frac{e^{2i\phi}}{e^{i\phi}+2}\bigg|=\frac{\big|e^{2i\phi}\big|}{\big|e^{i\phi}+2\big|}=\frac{1}{\big|e^{i\phi}+2\big|}$$

From there, I don't get how to determine the upper bound of the modulus of $f$ to find the maximum. Also, at which points does it occur?

Answer 1

Note that$$\bigl|f(z)\bigr|=\frac1{\sqrt{\bigl(\cos(\phi)+2\bigr)^2+\bigl(\sin(\phi)\bigr)^2}}=\frac1{\sqrt{5+4\cos(\phi)}}.$$It is clear that the maximum is $1$, attained when $\phi=\pi$.

Answer 2

$2+e^{i\phi} $ is $(x-2)^2+ y^2=1$ so that $|(x,y)|\geq |(1,0)|=1$