What is the maximum of this expression (function)?

[NaturePaper]

Hi,
Will anybody help me to find the maximum of the following expression:

|\cos^m\theta_1<br /> (c\cos^n\theta_2+se^{i(\gamma-n\lambda_2)}\sin^n\theta_2)+e^{-im\lambda_1}\sin^m\theta_1 (\pm ce^{-in\lambda_2}\sin^n\theta_2+se^{i\gamma}\cos^n\theta_2)|^2

where m,n\ge 2 are fixed positive integers; c,s\in (0,1) \mbox{ and } \gamma are fixed reals; and we have to maximize with respect to \theta_1, \theta_2,\lambda_1,\lambda_2 in the range 0\le\theta_1, \theta_2\le\frac{\pi}{2};~0\le\lambda_1,\lambda_2\le\pi.

My guess is the answer will be \max\{c^2, s^2\}. But I am unable to prove it (even I don't know if I am correct). Please help me.

 

[NaturePaper]

Here is my trying... Can anybody verify it, please(if there is any misconception):

We have

<br /> |\cos^m\theta_1<br /> (c\cos^n\theta_2+se^{i(\gamma-n\lambda_2)}\sin^n\theta_2)+e^{-im\lambda_1}\sin^m\theta_1 (\pm ce^{-in\lambda_2}\sin^n\theta_2+se^{i\gamma}\cos^n\theta_2)|


<br /> \le\cos^m\theta_1|<br /> (c\cos^n\theta_2+se^{i(\gamma-n\lambda_2)}\sin^n\theta_2)|+\sin^m\theta_1| (\pm ce^{-in\lambda_2}\sin^n\theta_2+se^{i\gamma}\cos^n\theta_2)|


<br /> \le \cos^m\theta_1(c\cos^n\theta_2+s\sin^n\theta_2)+\sin^m\theta_1 ( c\sin^n\theta_2+s\cos^n\theta_2)<br />

Now by calculus (I mean by differentiating w.r.t. \theta_1,\theta_2) the maximum of the r.h.s. of the last inequality is obtained as \max\{c,s\}.