On certain conditions for \(p\)-valence of analytic functions (Q1916573)

Let \(f\) be a function analytic and locally univalent in the unit disk \(\mathbb{D}\). The author establishes several sufficient conditions in terms of \(f''(z)/f'(z)\) under which \(f\) is at most \(p\)-valent in \(\mathbb{D}\). Justifications are based on earlier results of \textit{T. Umezawa} [Tohôku Math. J., II. Ser. 7, 212-228 (1955; Zbl 0066.32603)] and \textit{P. L. Shabalin} [Tr. Semin. Kraevym. Zadacham 16, 218-226 (1979; Zbl 0447.30010)].