This question is from Textbook Ponnusamy Silvermann page 283 , section application of Cauchy theorem.
Find maximum and minimum values of (a) $|5+2i z^2| $ on |z|$\leq$1. (b) f(z) =$ e^z$ on |z|=r
Attempt: maximum and minimum modulus theorem is to be used.
For (b) maximum at $e^r$ and minimum at $e^{-r}$ . Is it correct?
For (a) I am confused on how to use the principle. Maximum and minimum both will be at boundary. But how to find those values in this case.
Kindly help for part (a). Hope I am right in(b) .
If wrong kindly tell.
Observe that $|5+2iz^2| \leq 5+2|z|^2 \leq 7 $ for $|z| =1$ which is achieved at $z= \sqrt{-i}$ which lies on the boundary as $|z|^2= |-i|=1 \implies |z|=1.$ For minimum, use reverse triangle inequality and observe that $ |5+2iz^2| \geq 3$ on $|z|=1$ which is achieved at $z=i$.