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What is the maximum of this expression (function)?

  1. Apr 27, 2010 #1
    Hi,
    Will anybody help me to find the maximum of the following expression:

    [tex]|\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[/tex]

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

    My guess is the answer will be [tex]\max\{c^2, s^2\}[/tex]. 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. jcsd
  3. Apr 27, 2010 #2
    Here is my trying.... Can anybody verify it, please(if there is any misconception):

    We have

    [tex]
    |\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)|[/tex]


    [tex]
    \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)|[/tex]


    [tex]
    \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)
    [/tex]

    Now by calculus (I mean by differentiating w.r.t. [tex]\theta_1,\theta_2[/tex]) the maximum of the r.h.s. of the last inequality is obtained as [tex]\max\{c,s\}[/tex].
     
    Last edited: Apr 27, 2010
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