Find maximum and minimum argument of $f(z)=(z^4-1)/4$ where $z$ belongs to square with vertices $\{1,i,-1,-i\}$

Question

let z be a complex number with |z|<1, now how to find the max and min argument of $(z^4-1)/4$

This original problem given below comes from kheavan.files.wordpress.com/2011/10/mathematical-olympiads-1997-1998-problems-solutions-from-around-the-world-maa-problem-book-225p-b002kypabi.pdf

(question number 1 section 1.9)

Do read the solution to understand what I am really asking.

Let P be a point inside or on the sides of a square ABCD. Determine the minimum and maximum possible values of f(P) = ∠ABP + ∠BCP + ∠CDP + ∠DAP.

I want the answer to my question, not to the original problem

Answer 1

Your function

$$f(z)=(z^4-1)/4$$

can be considered as the composition of 3 functions/transforms :

$$z \to z^4=u \mapsto u-1=v \mapsto v/4$$

the two last transforms are a translation and a homothety ("shrinking" from the origin with ratio 4:1)

The closed unit disk $C((0,0),1)$ is sent onto itself by the first transform (each point being reached 4 times). Then it is transformed into C((-1,0),1), and finally into $C((-\tfrac14,0),\tfrac14)$ itself (each point being reached 4 times as can be understood by seing the second figure).

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Edit : I have had a look at the original question, with which I had some difficulty like you had. I understand now properly with the aid of a figure.

This figure represents the initial square with circumscribed unit circle ; and their images : no surprise for the image of the circle, but we couldn't truely await that the image of the square is a kind of droplet with limiting angles $3 \pi/4$ and $5 \pi/4$. All the vertices of the square are "mapped" on the apex of the drop : maybe you know that a complex transformation preserves the angles : the image of a $\pi/2$ angle remains a $\pi/2$ angle...

Fig. 1 : The image by $f$ 1) of the unit circle is the small circle 2) of the square is the shape looking like a drop. See Fig. 2.

Matlab program for the generation of this figure :

clear all;close all;hold on;axis equal
t=0:0.01:1;
f=@(z)((z.^4-1)/4),
for k=1:4
   z=(i^k)*(t+(1-t)*i);plot(z,'b');plot(f(z),'b'); % 4 square's sides
end;
z=exp(2*pi*i*t);plot(z,'r');plot(f(z),'r')

In order to understand the mapping, here is a complementary representation displaying arrows joining $z$ to $f(z)$ for different values of $z$, either at the boundary of a quarter of circle (green arrows) or of a side of the square (blue arrows); please note for example that the midpoint of the side (any side in fact) of the square is mapped onto the bottom of the drop.

Fig. 2 : Some examples of points $z$ and their resp. images $f(z)$ connected by an arrow. A quarter of the unit circle or a single side of the square are enough for defining the little circle and the drop.