# What is the maximum of this expression (function)?

1. Apr 27, 2010

### NaturePaper

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

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

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.

Last edited: Apr 27, 2010
2. Apr 27, 2010

### NaturePaper

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

We have

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

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

$$\le \cos^m\theta_1(c\cos^n\theta_2+s\sin^n\theta_2)+\sin^m\theta_1 ( c\sin^n\theta_2+s\cos^n\theta_2)$$

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 $$\max\{c,s\}$$.

Last edited: Apr 27, 2010