Authors’ summary: For any complex Banach space \(X\) and each \(p\in [1,\infty),\) we introduce the \(p\)-Bohr radius of order \(N\in \mathbb{N},\) \(\tilde{R}_{p,N}(X),\) defined by \[ \tilde{R}_{p,N}(X)=\sup \Bigl\{r\ge 0:\sum_{k=0}^N \|x_k\|^pr^{pk}\le \|f\|^p_{H^\infty(D,X)} \Bigr\}, \] where \(f(z)=\sum^\infty_{k=0}x_kz^k \in H^\infty(D,X).\) Here \(D=\{z\in \mathbb{C}:|z|<1\}\) 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 \(N\in \mathbb{N}\). For \(p\in[2,\infty),\) a complex Banach space \(X\) is called \(p\)-uniformly \(\mathbb{C}\)-convex of order \(N\) if there exists a constant \(\lambda>0\) such that \[ \big(\|x_0\|^p+\lambda\|x_1\|^p+\lambda^2\|x_2\|^p+\cdots+\lambda^N\|x_N\|^p\big)^{1/p}\leq\max_{\theta\in[0,2\pi)} \Bigl\| x_0+\sum_{k=1}^N e^{i\theta}x_k \Bigr\| \tag{1} \] for all \(x_0, x_1, \dots, x_N\in X.\) We denote \(A_{p,N}(X),\) the supremum of all such constants \(\lambda\) satisfying \((1).\) We obtain the lower and upper bounds of \(\tilde{R}_{p,N}(X)\) in terms of \(A_{p,N}(X).\) 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 C-convex of order \(N\) if, and only if, the \(p\)-Bohr radius of order \(N\) \(\tilde{R}_{p,N}(X)>0.\) We also study the \(p\)-Bohr radius of order \(N\) for the Lebesgue spaces \(L_q(\mu)\) for \(1\le p<q<\infty\) or \(1\le q\le p<2.\) Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk \(D\) into \(B(H),\) where \(B(H)\) denotes the space of all bounded linear operator on a complex Hilbert space \(H.\)