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On starlikeness of certain integral transforms. (English) Zbl 0757.30031

The author shows that if \(f\) is an analytic function of the unit disk with \[ \text{Re} f'(z)>-{{(2\ln 2-1)(3-2\ln 2)} \over {3-(2\ln 2-1)(3- 2\ln 2)}}\approx-0.262\dots, \] then \(F(z)=\int_ 0^ z (f(t)/t)dt\) is starlike univalent.
Further, if \(\text{Re} f'(z)>-\rho\approx-0.09\dots\), where \(\rho\) is given implicitly, then \(G(z)=(2/z)\int_ 0^ z f(t)dt\) is starlike univalent.
Reviewer: W.Koepf (Berlin)

MSC:

30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30C25 Covering theorems in conformal mapping theory
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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