Fournier, R.; Ponnusamy, S. A class of locally univalent functions defined by a differential inequality. (English) Zbl 1110.30008 Complex Var. Elliptic Equ. 52, No. 1, 1-8 (2007). Let \(\mu\in\mathbb C\) and \(\lambda\geq0\). The authors consider the class \(U(\lambda,\mu)\) which consists of functions \(f(z)=z+\dots\) analytic in the unit disk \(D\) and such that \[ \frac{f(z)}{z}\neq0\;\;\text{and}\;\;\left| f'(z)\left(\frac{z}{f(z)}\right)^{\mu+1}-1\right| \leq\lambda,\;\;z\in D. \] The main results are given in two theorems. Theorem 1: For \(\mu\in\mathbb C\), \(\text{Re}\mu<1\), all functions \(f\in U(\lambda,\mu)\) are starlike iff \[ 0\leq\lambda\leq\frac{| 1-\mu| }{\sqrt{| 1-\mu| ^2+| \mu| ^2}}. \] Theorem 2: For \(\mu\in\mathbb C\), \(\text{Re}\mu<1\), all functions \(f\in U(\lambda,\mu)\) are spirallike iff \[ 0\leq\lambda\leq\min\left(1,\frac{| 1-\mu| }{| \mu| }\right). \] Reviewer: Dmitri V. Prokhorov (Saratov) Cited in 2 ReviewsCited in 34 Documents MSC: 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) Keywords:differential inequality; starlike function; spirallike function PDFBibTeX XMLCite \textit{R. Fournier} and \textit{S. Ponnusamy}, Complex Var. Elliptic Equ. 52, No. 1, 1--8 (2007; Zbl 1110.30008) Full Text: DOI References: [1] Aksentiev LA, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika 3 pp 3– (1958) [2] Duren PL, Univalent Functions (1983) [3] DOI: 10.1080/17476938908814330 · Zbl 0639.30016 · doi:10.1080/17476938908814330 [4] DOI: 10.1080/0278107031000073614 · Zbl 1034.30008 · doi:10.1080/0278107031000073614 [5] DOI: 10.1007/BF01899211 · Zbl 0058.06302 · doi:10.1007/BF01899211 [6] DOI: 10.1007/BF01890578 · Zbl 0634.41007 · doi:10.1007/BF01890578 [7] DOI: 10.1090/S0002-9939-1972-0299773-3 · doi:10.1090/S0002-9939-1972-0299773-3 [8] Obradović M, Hokkaido Mathematical Journal 27 pp 329– (1998) · Zbl 0908.30009 · doi:10.14492/hokmj/1351001289 [9] Obradović M, Analysis 22 pp 225– (2002) · doi:10.1524/anly.2002.22.3.225 [10] DOI: 10.1007/BF02358538 · Zbl 0898.30019 · doi:10.1007/BF02358538 [11] DOI: 10.1017/S1446788700002482 · Zbl 0981.30010 · doi:10.1017/S1446788700002482 [12] Ruscheweyh St, Convolutions in Geometric Function Theory (1982) · Zbl 0499.30001 [13] DOI: 10.1093/qmath/23.2.135 · Zbl 0236.30019 · doi:10.1093/qmath/23.2.135 [14] Singh V, Indian Journal of Pure and Applied Mathematics 8 pp 1370– (1977) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.