Polylogarithms in the theory of univalent functions
Let \(a,b> -1,p\) and \(q\) nonzero real numbers such that \(p+q \geq 1\). In this paper, the authors study geometric mapping properties of the (normalized) generalized polylogarithm defined in the unit disk \(\Delta = \{z\in\mathbb{C}: |z |< 1\}\) by \[ \Phi_{p,q} (a,b; z) = \sum^\infty_{n=1} {(1+ a)^p(1+ b)^q \over (n+ a)^p(n+ b)^q} z^n = z + \dots. \] Under various parameter restrictions the authors prove several results concerning the membership of the function \(\Phi_{p,q} (a,b;z)\) in the well-known subclasses of univalent functions such as the class \(K_1 \cap S^*\).
scientific article; zbMATH DE number 933262