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Starlikeness of integral transforms and duality. (English) Zbl 1244.30008

Let \(D\) be the complex unit disc and \(\mathcal A\) the class of functions \(f(z)=z+a_2z^2+\cdots\). For \(f\in\mathcal{A}\) also satisfing the condition \[ \mathrm{Re} e^{\mathrm{i}\phi}\left((1-\alpha+2\gamma)\frac{f(z)}{z}+(\alpha-2\gamma)f'(z)+\gamma zf'' (z)-\beta\right)> 0 \] for suitable \(\phi\), \(\alpha\), \(\beta\) and \(\gamma\), the authors give sufficient conditions so that the function defined by \[ V_{\lambda}(f)(z)=\int_0^z\lambda(t)\frac{f(tz)}{t}dt \] (with \(\lambda\) chosen so that the above formula generalizes some results of other authors, but also provide new results) is starlike. Particular cases of \(\lambda\) are taken into account. Some consequences are also given. One of them gives a sharp estimate for the real constant \(\beta<1\) that ensures starlikeness of a function \(f\in\mathcal A\) that satisfies the condition \(\mathrm{Re}(f'(z)+\alpha zf'' (z)+\gamma z^2f'''(z))>\beta\).

MSC:

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