  Applying a linearization theorem due to J. Mujica, we study the ideals of
bounded holomorphic mappings $\mathcal{H}^\infty\circ\mathcal{I}$ generated by
composition with an operator ideal $\mathcal{I}$. The bounded-holomorphic dual
ideal of $\mathcal{I}$ is introduced and its elements are characterized as
those that admit a factorization through $\mathcal{I}^\mathrm{dual}$. For
complex Banach spaces $E$ and $F$, we also analyze new ideals of bounded
holomorphic mappings from an open subset $U\subseteq E$ to $F$ such as
$p$-integral holomorphic mappings and $p$-nuclear holomorphic mappings with
$1\leq p<\infty$. We prove that every $p$-integral ($p$-nuclear) holomorphic
mapping from $U$ to $F$ has relatively weakly compact (compact) range.
