arXiv subjects
Publications and source records attributed to K. Chandrasekran.
The main objective of the present article is to make interconnection between the Generalized Hyergeometric series and some subclasses of normalized analytic functions with positive(Tailor's) coefficients in the open unit disc $\mathbb{D} =\{z:\, |z|<1\}$.
In this work, we derived the necessary and sufficient conditions on parameters for $_3F_2(^{a,b,c}_{b+1,c+1};z)$ Hypergeometric Function to be in the classes $\mathcal{M}^{\ast}(λ,α)$ and $\mathcal{N}^{\ast}(λ,α)$ and information regarding the image of function $_3F_2(^{a,b,c}_{b+1,c+1};z)$ belonging to $\mathcal{R}^τ(A,B)$ by applying the convolution operator in open unit disc $\mathbb{D} =\{z:\, |z|<1\}$.
In this article, Using Hadamard product for $_4F_3\left(^{a_1,\, a_2,\, a_3,\, a_4}_{b_1,\, b_2,\, b_3};z\right)$ hypergeometric function with normalized analytic functions in the open unit disc, an operator $\mathcal{I}^{a_1,a_2,a_3,a_4}_{b_1,b_2,b_3}(f)(z)$ is introduced. Geometric properties of $_4F_3\left(^{a_1,\, a_2,\, a_3,\, a_4}_{b_1,\, b_2,\, b_3};z\right)$ hypergeometric functions are discussed for various subclasses of univalent functions. Also, we consider an operator $\mathcal{I}^{ a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4} }_{ \frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4} }(f)(z)$$= z\, _5F_4\left(^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}}_{\frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4}}; z\right)*f(z)$, where, $_5F_4(z)$ hypergeometric function and the $*$ is usual Hadamard product. In the main results, conditions are determined on $ a,b,$ and $c$ such that the function $z\, _5F_4\left(^{a,\frac{b}{4},\frac{b+1}{4},\frac{b+2}{4},\frac{b+3}{4}}_{\frac{c}{4}, \frac{c+1}{4}, \frac{c+2}{4},\frac{c+3}{4}}; z\right)$ is in the each of the classes $ \mathcal{S}^{*}_λ $, $ \mathcal{C}_λ$, $UCV$ and $\mathcal{S}_p$. Subsequently, conditions on $a,\,b,\,c,\, λ,$ and $β$ are determined using the integral operator such that functions belonging to $\mathcal{R}(β)$ and $\mathcal{S}$ are mapped onto each of the classes $\mathcal{S}^*_λ$, $\mathcal{C}_λ$, $UCV$, and $\mathcal{S}_p$.
We consider the integral operator $\mathcal{I}^{a,b,c}_{d,e}(f)(z)$ involving Clausen's Hypergeometric Function by means of convolution introduced by Chandrasekran and Prabhakaran for investigation. The conditions on the parameters $ a,b, c$ are determined using the integral operator $\mathcal{I}^{a,\frac{b}{2},\frac{b+1}{2}}_{\frac{c}{2}, \frac{c+1}{2}}(f)(z)$ to study the geometric properties of Clausen's Hypergeometric Function for various subclasses of univalent functions.
In this paper, we obtain various conditions on the parameters $a,\, b,\, c\,, d$ and $e$ for which the hypergeometric functions $z\,_3F_2(a,b,c;d,e;z)$ to be in the class of all close-to-convex function with respect to some well known convex functions.