In this paper, for complex Banach spaces E, F and \(1\le p\le \infty \), the subspaces \({\mathcal {H}}_{p}^{\gamma }(E,F)\) of the space \({\mathcal {H}}_{b}(E,F)\) consisting of holomorphic mappings of bounded type from E into F, have been introduced and studied. Here the notation \(\gamma \) stands for a comparison function \(\gamma \) which is an entire function defined on the complex plane, as \(\gamma (z)=\sum \nolimits _{n=0}^{\infty } \gamma _{n} z^{n}, \gamma _{n} >0\) for each \(n \in {\mathbb {N}}_{0}\) with \(\gamma _{n}^{\frac{1}{n}}\rightarrow 0\) and \(\frac{\gamma _{n+1}}{\gamma _{n}} \downarrow 0\) as n increases to \(\infty \). Besides considering the relationships amongst these spaces, their vector valued sequential analogues have also been obtained for \(1\le p <\infty \). These results are used in obtaining the dual and Schauder decomposition of \({\mathcal {H}}_{p}^{\gamma }(E,F)\), \(1\le p <\infty \). The continuity of differentiation and translation operator has been proved by restricting \(\gamma \) suitably and the spectrum of the differentiation operator \(D_a\) has been investigated. Finally, the continuity and compactness of the composition operator \(C_{\phi }\), defined corresponding to a holomorphic function \(\phi \) have been investigated.