Summary: This paper aims to investigate characterizations on parameters \(k_{1}\), \(k_{2}\), \(k_{3}\), \(k_{4}\), \(k_{5}\), \(l_{1}\), \(l_{2}\), \(l_{3}\), and \(l_{4}\) to find relation between the class of \(\mathcal{H}(k, l, m, n, o)\) hypergeometric functions defined by \[ _{5}F_{4} \left[ \begin{matrix} k_{1}, k_{2}, k_{3}, k_{4}, k_{5} \\ l_{1}, l_{2}, l_{3}, l_{4} \end{matrix} : {z} \right] = \sum^{\infty}_{n = 2} \frac{(k_{1})_{n}, (k_{2})_{n}, (k_{3})_{n}, (k_{4n},(k_{5})_{n}}{(l_{1})_{n}, (l_{2})_{n}, (l_{3})_{n}, (l_{4})_{n}} z^{n}. \] We need to find \(k\), \(l\), \(m\) and \(n\) that lead to the necessary and sufficient condition for the function \(zF([W])\), \(G = z(2 - F([W]))\) and \(H_{1} [W] = z^{2} \frac{d}{dz} (ln (z) - h(z))\) to be in \(\mathcal{S}^{\ast} (2^{-r})\), \(r\) is a positive integer in the open unit disc \(\mathcal{D} = \{z : |z| < 1, z \in \mathbb{C}\}\) with \[ h(z) = \sum^{\infty}_{n = 0} \frac{(k)_{n}(l)_{n}(m)_{n}(n)_{n}(1 + \frac{k}{2})_{n}}{(\frac{k}{2})_{n}(1 + k - 1)_{n}(1 + k - m)_{n}(1 + k - n)_{n} n(1)_{n}} z^{n} \] and \[ [W] = \left[ \begin{matrix} k, 1, + \frac{k}{2}, l, m, n \\ \frac{k}{2}, 1 + k - l, 1 + k - m, 1 + k - n \end{matrix} : {z} \right]. \]