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

Let $D$ be the complex unit disc and $𝒜$ the class of functions $f\left(z\right)=z+{a}_{2}{z}^{2}+\cdots$. For $f\in 𝒜$ also satisfing the condition

$\mathrm{Re}{e}^{\mathrm{i}\varphi }\left(\left(1-\alpha +2\gamma \right)\frac{f\left(z\right)}{z}+\left(\alpha -2\gamma \right){f}^{\text{'}}\left(z\right)+\gamma z{f}^{\text{'}\text{'}}\left(z\right)-\beta \right)>0$

for suitable $\varphi$, $\alpha$, $\beta$ and $\gamma$, the authors give sufficient conditions so that the function defined by

${V}_{\lambda }\left(f\right)\left(z\right)={\int }_{0}^{z}\lambda \left(t\right)\frac{f\left(tz\right)}{t}dt$

(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 $f\in 𝒜$ that satisfies the condition $\mathrm{Re}\left({f}^{\text{'}}\left(z\right)+\alpha z{f}^{\text{'}\text{'}}\left(z\right)+\gamma {z}^{2}{f}^{\text{'}\text{'}\text{'}}\left(z\right)\right)>\beta$.

##### MSC:
 30C45 Special classes of univalent and multivalent functions
##### References:
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