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Zbl 1244.30008
Ali, Rosihan M.; Badghaish, Abeer O.; Ravichandran, V.; Swaminathan, A.
Starlikeness of integral transforms and duality. (English)
[J] J. Math. Anal. Appl. 385, No. 2, 808-822 (2012). ISSN 0022-247X

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 $\mathrm{Re}(f'(z)+\alpha zf'' (z)+\gamma z^2f'''(z))>\beta$.
[Eugen Drăghici (Sibiu)]

MSC 2000:: *30C45 Special classes of univalent and multivalent functions

Keywords: duality; convolution; univalence; starlike function; integral transform

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Zentralblatt MATH Berlin [Germany]