TITLE: general method for finding minimum value of complex function
QUESTION [0 upvotes]: I know that we can find the maximum value of entire functions by maximum modulus principle, is there any general method to find the minimum value of complex functions.
For $ f(z) = z^2+2 .$ what will be the minimum value of $ \vert f(z)\vert $ in the unit disc?
I think it should be at $z=0$ but I am not sure.

REPLY [0 votes]: Go by taking let Z = ( Cos x + i Sin x ) ; 
i.e f(Z) = Cos 2x + 2 + i Sin 2x $\\$
|f(Z)| = $ \sqrt{ (2+Cos\,2x)^2 + (Sin\,2x)^2} $ $\\$
|f(Z)| = $\sqrt{5\;+4Cos\,2x} $ = 1 $\\$
When Cos 2x = -1.
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