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Math Help - connected subharmonic function

  1. #1
    Newbie
    Joined
    May 2011
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    5

    Lightbulb connected subharmonic function

    Hi there
    i need some help with this question

    let D=D(0,1),let u be a continuous function on (\bar{D}) that is subharmonic on D,and let E={z \in \bar{D} :u(z)\leqslant 0}
    Explain why C\E is connected,

    and let u be subharmonic function on the strip U={z in C : -1< Im z < 1}
    such that
    lim sup u(z)\v(z)\leqslant 0 (w \in \partial \infty U)
    where v(x+iy)=coshx cosy

    show that u\leqslant 0

    thanks
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  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    For the first use the maximum principle to see that there can be no bounded connected component, and since E is compact the complement has precisely one unbounded component.

    The second one I don't understand, what does
    lim sup u(z)\v(z)\leqslant 0 (w \in \partial \infty U)
    mean
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