Univalent holomorphic mappings on a complex manifold with a \(C^1\) exhaustion function (Q1296397)

By introducing the notion of a \(\Phi\)-like domain, \textit{L. Brickman} [Bull. Am. Math. Soc. 79, 555-558 (1973; Zbl 0273.30010)] and \textit{K. R. Gurganus} [Trans. Am. Math. Soc. 205, 389-406 (1975; Zbl 0299.32018)] generalized the concept of starlike and spirallikeness in one and several complex variables. They also gave an analytic condition for a normalized holomorphic mapping of the unit ball to be univalent and to have \(\Phi\)-like image. In this paper, the author studies \(\Phi\)-like domains in \(\mathbb C^n\) that are images of holomorphic functions \(f\) defined on a complex manifold \(M\) having an exhaustion function. The existence of the exhaustion function enables the author to derive a sufficient condition for the image \(f(M)\) to be \(\Phi\)-like. In addition, and using arguments based on subordination, he establishes necessary and sufficient conditions for the univalence of such mappings \(f\).