First let us define analytic capacity: Let $\displaystyle E\in \mathbb{C}$ is compact, let $\displaystyle U_E$ be the collection of functions f holomorphic on $\displaystyle \hat{\mathbb{C}} - E$ ($\displaystyle \hat{\mathbb{C}}$ is $\displaystyle \mathbb{C} \cup\infty$) and $\displaystyle f(\infty) = 0$ and $\displaystyle \sup{|f(z)|}\leq 1$; the analytic capacity of E is defined as :
$\displaystyle \gamma (E) = \sup_{f\in U_E}{f'(\infty)}$.
If $\displaystyle E_1$, $\displaystyle E_2$ are disjoint compact sets which can be separated by a line $\displaystyle l$ (i.e. $\displaystyle l\cap E_j = \emptyset, j = 1,2$), then show that:
$\displaystyle \gamma(E_1\cup E_2)\leq \gamma(E_1) + \gamma(E_2)$ .