Geometric Properties of Generalized Hypergeometric Functions

In this article, Using Hadamard product for ${}_4{F}_3({}_{b_1,b_2,b_3}^{a_1,a_2,a_3,a_4};z)$ hypergeometric function with normalized analytic functions in the open unit disc, an operator ${\mathcal{I}}_{b_1,b_2,b_3}^{a_1,a_2,a_3,a_4}(f)(z)$ is introduced. Geometric properties of ${}_4{F}_3({}_{b_1,b_2,b_3}^{a_1,a_2,a_3,a_4};z)$ hypergeometric functions are discussed for various subclasses of univalent functions. Also, we consider an operator ${\mathcal{I}}_{\frac{c}{4},\frac{c+1}{4},\frac{c+2}{4},\frac{c+3}{4}}^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}}(f)(z)$$=z{}_5{F}_4({}_{\frac{c}{4},\frac{c+1}{4},\frac{c+2}{4},\frac{c+3}{4}}^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}};z)\ast f(z)$, where, ${}_5{F}_4(z)$ hypergeometric function and the $\ast$ is usual Hadamard product. In the main results, conditions are determined on $a,b,$ and $c$ such that the function $z{}_5{F}_4({}_{\frac{c}{4},\frac{c+1}{4},\frac{c+2}{4},\frac{c+3}{4}}^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}};z)$ is in the each of the classes ${\mathcal{S}}_{\lambda }^{\ast }$, ${\mathcal{C}}_{\lambda }$, $UCV$ and ${\mathcal{S}}_p$. Subsequently, conditions on $a,b,c,\lambda ,$ and $\beta$ are determined using the integral operator such that functions belonging to $\mathcal{R}(\beta )$ and $\mathcal{S}$ are mapped onto each of the classes ${\mathcal{S}}_{\lambda }^{\ast }$, ${\mathcal{C}}_{\lambda }$, $UCV$, and ${\mathcal{S}}_p$.