Convolution properties of \(p\)-valent functions associated with a generalization of the Srivastava-Attiya operator
Summary: Let \(\mathcal A_p\) denote the class of functions analytic in the open unit disc \(\mathbb U\) and given by the series \(f(z) = z^p + \sum^\infty_{n=p+1}a_nz^n\). For \(f \in \mathcal A_p\), the transformation \[ \mathcal I^\lambda_{p, \delta} : \mathcal A_p \to \mathcal A_p \] defined by \[ \mathcal I^\lambda_{p, \delta}f(z) = z^p + \sum^\infty_{n=p+1}((p + \delta)/(n + \delta))^\lambda a_nz^n, \] where \(\delta + p \in \mathbb C \setminus \mathbb Z^-_0\), \(\lambda \in \mathbb C\), \(z \in \mathbb U\), has been recently studied as fractional differintegral operator by \textit{A. K. Mishra} and the author [Complex Var. Elliptic Equ. 55, No. 7, 677--689 (2010; Zbl 1195.30028)]. In the present paper, we observe that \(\mathcal I^\lambda_{p, \delta}\) can also be viewed as a generalization of the Srivastava-Attiya operator. Convolution preserving properties for a class of multivalent analytic functions involving an adaptation of the popular Srivastava-Attiya transform are investigated.
scientific article; zbMATH DE number 6191256
scientific article