In the paper we give some estimates of the determinant of the second order Hankel functional matrix \(H_{2}^{1}=\left[ \begin{array}{cc} Q_{f,1} & Q_{f,2}\\ Q_{f,2} & Q_{f,3} \end{array} \right] \) , where \(Q_{f,1},Q_{f,2},Q_{f,3}\) are the initial terms in a homogeneous polynomial series expansions of functions from two subfamilies of the family \(\mathcal {H}ol(\mathbb {B},\mathbb {C})\) of all functions \(f:\mathbb {B}\rightarrow \mathbb {C},\) holomorphic on the open unit ball \(\mathbb {B}\) of a Banach space \(\mathbb {X}.\) As application of one of such estimates, we prove a sharp estimate of the determinant of the second order Hankel scalar matrix \(H_{2}^{2}=\left[ \begin{array}{cc} T_{x}(Q_{F,2}(x)) & T_{x}(Q_{F,3}(x))\\ T_{x}(Q_{F,3}(x)) & T_{x}(Q_{F,4}(x)) \end{array} \right] .\) Here x are the points from \(\mathbb {B}\diagdown \{0\},\) the mappings \(Q_{F,2},Q_{F,3},Q_{F,4}\) are the successive terms in a homogeneous polynomial series expansions of normalized starlike mappings \(F:\mathbb {B}\rightarrow \mathbb {X}\) and \(T_{x}-\) elements of the dual space \(\mathbb {X}^{*}\) such that \(\left\| T_{x}\right\| =1,T_{x}(x)=||x||.\)
