The duals to the spaces of analytic vectorvalued functions and the duality of functions, generated by these spaces

Let $X$ a complex Banach space, $1<p<\infty$, $1/p+1/p'=1$; $A^p(X)$ is the space of all $X$-valued analytic in the open disk $L^p$-integrable functions. By means of the natural duality it is proved that $A^p(X)^*=A^{p'}(X^*)$. Let $\mathbf A^p$ and $\mathbf H^p$ be the functors in a category of Banach spaces, generated by $A^p(X)$ and the Hardy space $H^p(X)$ respectively. With some restrictions on the category the following it true: 1) $D\mathbf A^p=\mathbf A^{p'}$; 2) $H^p(X)^*=D\mathbf H^p(X^*)$; 3) $D\mathbf H^p\ne\mathbf H^{p'}$ in the category of all separable reflexive Banach spaces; 4) the functors $\mathbf A^p$ and $\mathbf H^p$ are reflexive.