On subordinations for certain analytic functions associated with Noor integral operator
Let \(\mathcal{A}(p)\) be the class of functions of the form \[ f(z)=z^{p}+\sum_{k=p+1}^{\infty}a_{k}z^{k}\quad (p\in N=\left\{1,2,3,\ldots\right\}), \] which are analytic in the open unit disc \(U=\left\{z\in \mathbb{C}\;,| z| <1\right\}\). Let \(f_{n,p}(z)=z^{p}/(1-z)^{n+p}\;(n>-p)\) and let \(f_{n,p}^{(+)}(z)\) be defined such that \[ f_{n,p}(z)\ast f_{n,p}^{(+)}(z)=\frac{z^p}{(1-z)^{p+1}}, \] where the symbol \((\ast)\) stands for the Hadamard product. Consider the integral operator \(I_{n,p}:\mathcal{A}(p)\rightarrow\mathcal{A}(p)\) defined by \[ I_{n,p}f(z)=f_{n,p}^{(+)}(z)\ast f(z)= \left(\frac{z^{p}}{(1-z)^{n+p}}\right)^{(+)}\ast f(z)\quad (f\in\mathcal{A}(p)). \] The operator \(I_{n,p}f(z)\) is called the Noor integral operator of \((n+p-1)\)th order of f. In the paper the authors use the method of differential subordinations to obtain some properties of the Noor integral operator \(I_{n,p}f(z)\).
scientific article; zbMATH DE number 5163824