On subadditivity of analytic capacity for two continua (Q799833)

Let \(\gamma\) (K) denote the analytic capacity of the compact plane set K. It is an important open problem whether \(\gamma\) is subadditive or even if \(\gamma (K_ 1\cup K_ 2)\leq M(\gamma (K_ 1)+\gamma (K_ 2))\) for some absolute constant M. The author shows that \(\gamma (K_ 1\cup K_ 2)\leq\gamma (K_ 1)+\gamma (K_ 2)\) holds if \(K_ 1\) and \(K_ 2\) are connected. The proof uses the conformal mapping of the complement of \(K_ 1\cup K_ 2\) onto the complement of two real slits.