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.
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.