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Using Jensen's Inequality

  1. Apr 4, 2006 #1
    Let [itex]x \in \mathbb{R}^n[/itex] and

    [tex]u_0>0, \qquad \int\limits_\Omega u_0(x) dx =1, \qquad E(t)=\int\limits_\Omega u(x,t)u_0(x)dx[/tex]

    Im having trouble proving the following inequality

    [tex]\int\limits_\Omega \frac{u_0(x)}{(1+u(x,t))^2}dx \ge \dfrac{1}{(1+E)^2}. \qquad \hbox{(1)}[/tex]

    I know i have to use Jensen's inequality

    [tex] f\left(\frac{1}{|\Omega|}\int\limits_\Omega u dx \right) \le \frac{1}{|\Omega|}\int\limits_\Omega f(u) dx [/tex],

    where [itex]f(u)[/itex] is convex.

    But in order to use it to prove (1), I need to rewrite the left hand side of the equation or use a previous inequality right?

    There is where im stuck. Can anybody give me a sugestion pls?
     
    Last edited: Apr 4, 2006
  2. jcsd
  3. Apr 4, 2006 #2
    Is it just me or nobody can see the TeX?
     
  4. Apr 4, 2006 #3
    I cannot either.
     
  5. Apr 6, 2006 #4
    Well, first of all, it would be nice if someone tell me why the TeX doesnt work. Second of all, i got it, so nevermind.

    [tex]\int[/tex]

    [tex]\Omega[/tex]

    [tex]\omega[/tex]

    no \int???? nice.....
     
    Last edited: Apr 6, 2006
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