# Setting up bounds for the integral $(\int_{U}|f(x,y)|^{2}dxdy)^{\frac{1}{2}}$?

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 $(1.2)$

$(1.2)$

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

• 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). Jan 24, 2018 at 18:46
• Thanks for the hint @Thomas :>). Jan 24, 2018 at 21:03
• @Thomas would you mind writing that as an answer? Jul 9, 2018 at 16:34
• @barto no -- if it helps.. Jul 9, 2018 at 20:32

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.