## 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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### References:

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