  We describe the strong dual space $({\mathcal O}^s (D))^*$ for the space
${\mathcal O}^s (D) =
  H^s (D) \cap {\mathcal O} (D)$ of holomorphic functions from the Sobolev
space $H^s(D)$, $s \in \mathbb Z$, over a bounded simply connected plane domain
$D$ with infinitely differential boundary $\partial D$. We identify the dual
space with the space of holomorhic functions on ${\mathbb C}^n\setminus
\overline D$ that belong to $H^{1-s} (G\setminus \overline D)$ for any bounded
domain $G$, containing the compact $\overline D$, and vanish at the infinity.
As a corollary, we obtain a description of the strong dual space $({\mathcal
O}_F (D))^*$ for the space ${\mathcal O}_F (D)$ of holomorphic functions of
finite order of growth in $D$ (here, ${\mathcal O}_F (D)$ is endowed with the
inductive limit topology with respect to the family of spaces ${\mathcal O}^s
(D)$, $s \in \mathbb Z$).
  In this way we extend the classical Grothendieck-K{\"o}the-Sebasti\~{a}o e
Silva duality for the space of holomorphic functions.
