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A class of locally univalent functions defined by a differential inequality. (English) Zbl 1110.30008

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 \[ 0\leq\lambda\leq\min\left(1,\frac{| 1-\mu| }{| \mu| }\right). \]

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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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)
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