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Oya Mert,Alaattin Akyar,İsmet Yildiz 호남수학회 2022 호남수학학술지 Vol.44 No.1
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 $$ {}_5 F_4\bigg [ \begin{matrix}k_{1},k_{2},k_{3},k_{4},k_{5}\\l_{1},l_{2},l_{3},l_{4}\end{matrix} :z\bigg ]=\sum_{n=2}^{\infty}\frac{(k_{1})_{n} (k_{2})_{n}(k_{3})_{n}(k_{4})_{n}(k_{5})_{n}}{(l_{1})_{n}(l_{2})_{n}(l_{3})_{n}(l_{4})_{n}(1)_{n}}z^{n}. $$ 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 $H_{1}[W]=z^{2}\dfrac{d}{dz}(ln(z)-h(z))$ to be in $\mathcal{S^{*}}(2^{-r})$,~~$r$ is a positive integer in the open unit disc $\mathcal{D}=\{z:|z|<1,z\in\mathbb{C}\}$ with $$ \displaystyle\ h(z)=\sum_{n=0}^{\infty}\frac{(k)_{n}(l)_{n}(m)_{n}(n)_{n}(1+\frac{k}{2})_{n}}{(\frac{k}{2})_{n}(1+k-l)_{n} (1+k-m)_{n}(1+k-n)_{n}n(1)_{n}}z^n $$ and $$ \displaystyle [W]=\bigg [\begin{matrix}k,1+\frac{k}{2},l,m,n\\\frac{k}{2},1+k-l,1+k-m,1+k-n\end{matrix} :z\bigg ]. $$