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In the text "Functions of a Complex Variable" by Robert E.Grenne and Steven G. Knartz I'm having the trouble with figuring out a method of attack for $\text{Proposition (1.1)}$ specifically getting the integral $(\int_{U}|f(x,y)|dxdy)^{\frac{1}{2}}$ into a more manageable form, may I have a hint to achieve this ?

$\text{Proposition (1.1)}$

Let $U \subset \mathbb{C}$ be an open set and let $K$ be a compact subset $U$. Show that there is a constant $C$ $\text{(depending on U and K)}$ such that if $f$ is holomorphic on $U$, then in $(1.2)$

$(1.2)$

$$\sup_{K}|f| \leq C \cdot \big(\int_{U}|f(x,y)|^{2}dxdy \big)^{\frac{1}{2}}$$

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    $\begingroup$ An answer for the case of a disc contained in a bigger disc can be found here: math.stackexchange.com/questions/1021451/…. You can use geometric argument to carry this over to your case (cover $K$ by discs of sufficiently small radius and choose a finite subcover). $\endgroup$
    – Thomas
    Jan 24, 2018 at 18:46
  • $\begingroup$ Thanks for the hint @Thomas :>). $\endgroup$
    – Zophikel
    Jan 24, 2018 at 21:03
  • $\begingroup$ @Thomas would you mind writing that as an answer? $\endgroup$ Jul 9, 2018 at 16:34
  • $\begingroup$ @barto no -- if it helps.. $\endgroup$
    – Thomas
    Jul 9, 2018 at 20:32

1 Answer 1

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An answer for the case of a disc contained in a bigger disc can be found here.

You can use a geometric argument to carry this over to your case. To this end,cover $K$ by discs of sufficiently small radius and then choose a finite subcover.

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