A class of integral operators preserving subordination and superordination for analytic functions
Define a family \(I^{h}_{\beta, \gamma}\) of integral operators as follows: \[ \displaystyle{I^{h}_{\beta, \gamma}f(z)=\left[\frac{ \gamma+ \beta}{z^{\gamma}} \int^{z}_{0} f^{\beta}(t)h^{\gamma -1}(t)h^{\prime}(t)dt \right]^{1/ \beta}}, \tag{1} \] where each of the functions \(f\) and \(h\) belongs to the class \(A\) and the parameters \(\beta \in \mathbb{C} \backslash \{0\}\), \(\gamma \in \mathbb{C}\), \(\mathrm{Re}(\gamma+\beta)>0\), are so constrained that the integral operators in (1) exist. Throughout this paper, the authors denote by \(\mathbf{A}_{\beta, \gamma}\) the following analytic function class: \[ \displaystyle{\mathbf{A}_{\beta, \gamma}=\left\{f\in \mathbf{H} (\mathbf{U}): \frac{f(z)}{z}\neq 0, \frac{I_{\beta, \gamma}(z)}{z}\neq 0, \,z\in \mathbb{U} \right\}}. \] In the present paper, the authors obtain the subordination and superordination preserving properties of the integral operator \(I^{h}_{\beta, \gamma}\) defined by \((1)\) with a sandwich-type theorem. They also consider an interesting application of the main results to the Gauss hypergeometric function.
scientific article; zbMATH DE number 6102611
scientific article