I'm looking for the maximum value of the modulus of a holomorphic function, and I am getting a bit stuck.
The function is $(z-1)\left(z+\frac{1}{2}\right)$ with domain $\,|z| \leq 1\,$
Now, I know by the maximum modulus principle the max value will occur on the boundary. So by multiplying the two expressions I get: $\left|z^2 - \frac{1}{2}z - \frac{1}{2}\right|$
writing in complex polar form (and applying MMP, so $\,r = 1\,$) I then get: $\left|e^{2i\theta} -\frac{1}{2}\,e^{i\theta}-\frac{1}{2}\right|$
And... this is where I am stuck. So any help would be greatly appreciated!