A Maxima problem

1. Sep 2, 2009

NaturePaper

Let

$$F=z_1\cos\theta_1\cos\theta_2+z_2(e^{i\alpha_1}\sin\theta_1\cos\theta_2+e^{i\alpha_2}\cos\theta_1\sin\theta_2)+z_3e^{i(\alpha_1+\alpha_2)}\sin\theta_1\sin\theta_2$$

where $$0\le\alpha_i\le\pi,~0\le\theta_i\le\pi/2$$ and $$z_i$$
are some fixed complex numbers.

Then how to find
$$\max_{\theta_i, \alpha_i}|F|$$

We note that $$|F|\le1$$.

Particularly, I want to know if there is any set of constraints like the case of optimization over real variables. [I know definitely that the conditions are $$\theta_1=\theta_2; \alpha_1=\alpha_2$$. But I have to establish it. So, how to prove it?]

2. Sep 4, 2009

NaturePaper

Some corrections, $$|F|\le1$$
should be $$|F|\le N \mbox{ i.e., |F| is bounded}$$.

I was trying to use the observation

$$\max|\sum z_i|=\sum|z_i|$$
occurs iff $$z_i$$s have equal argument. But for my case, since $$z_i$$s
are arbitrary, I can't drive out some common phase to get the maximum as $$\sum|z_i|$$. So. for generic $$z_i,~~|F|$$ should depend on $$\alpha_i$$s.

In light of these observations, my precise questions are:

1. Can we choose $$\theta_1=\theta_2$$ to get max|F|?
2. Can we choose $$\alpha_1=\alpha_2$$ too ?
3. Is the conditions 1. and 2. are necessary to get |F|?
4. How?