  For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce
the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$
defined by $$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0:
\sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D},
X)}\right\}, $$ where $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in
H^{\infty}(\mathbb{D}, X)$. Here $\mathbb{D}= \{z\in \mathbb{C}: |z| <1\}$
denotes the unit disk. We also introduce the following geometric notion of
$p$-uniformly $\mathbb{C}$-convexity of order $N$ for a complex Banach space
$X$ for some $N \in \mathbb{N}$. In this paper, for $p\in [2,\infty)$ and each
$N \in \mathbb{N}$, we prove that a complex Banach space $X$ is $p$-uniformly
$\mathbb{C}$-convex of order $N$ if, and only if, the $p$-Bohr radius of order
$N$ $\widetilde{R}_{p,N}(X)>0$. We also study the $p$-Bohr radius of order $N$
for the Lebesgue spaces $L^q (\mu)$ for $1\leq p<q<\infty$ or $1\leq q \leq p
<2$. Finally, we prove an operator valued analogue of a refined version of Bohr
and Rogosinski inequality for bounded holomorphic functions from the unit disk
$\mathbb{D}$ into $\mathcal{B(\mathcal{H})}$, where $\mathcal{B(\mathcal{H})}$
denotes the space of all bounded linear operator on a complex Hilbert space
$\mathcal{H}$.
