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What is the relation between $$\limsup_{r\to\infty}\log|f(re^{it})|$$ and $$\limsup_{|z|\to\infty}\log|f(z)|$$ where $z=re^{it}$, $r>0, 0.

I know that the first one is a function of $t$, but the second one is a constant (assuming both limits exist), I'm told that I have this relation but I don't know why?? any help

$$\limsup_{r\to\infty}\log|f(re^{it})|\leq \limsup_{|z|\to\infty}\log|f(z)|$$

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    The latter is the supremum over $t$ of the former. In particular, the value of the latter is at least the value of the former at any given $t$.2012-04-11
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    Yes, that's right, but why this is true! $$\max_{t}\limsup_{r\to\infty}\log|f(re^{it})|= \limsup_{|z|\to\infty}\log|f(z)|$$2012-04-11
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    So, any comments!!?2012-04-11

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