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Zbl 1146.30009
Ponnusamy, S.; Rønning, F.
Integral transforms of a class of analytic functions.
(English)
[J] Complex Var. Elliptic Equ. 53, No. 5, 423-434 (2008). ISSN 1747-6933; ISSN 1747-6941/e

Let $\Delta$ be the complex unit disc and $\mathcal A$ be the class of all analytic functions in $\Delta$, normalized with the conditions $f(0)=f'(0)-1=0$. A function in $\mathcal A$ is said to be in the class $\mathcal P_{\lambda}(\beta)$ if $\mathrm{Re}[\mathrm{e}^{\mathrm{i}\phi}(f'(z)+\gamma zf''(z)-\beta]>0$ in $\Delta$ ($\phi\in\mathbb{R}$, $\gamma\geq 0$ and $\beta<1$). For a nonnegative real-valued integrable function $\lambda(t)$ satisfying the normalizing condition $\int_0^1\lambda(t)dt=1$ and $f\in\mathcal A$ let $$F(z)=V_{\lambda}(f)(z)=\int_0^1\lambda(t)\frac{f(tz)}{t}dt$$ and $$\Lambda_{\gamma}(t)=\int_t^1\frac{\lambda(s)}{s^{1/\gamma}} ds,\;\;\gamma>0$$ $$\Pi_{\gamma}(t)=\int_t^1\Lambda_{\gamma}(s)s^{1/\gamma-2}ds \text{ for }gamma>0\text{ and } \Pi_{\gamma}(t)=\int_t^1\frac{\lambda(s)}{s}ds\text{ for }\gamma=0$$ If $$\frac{\beta}{1-\beta}=-\int_0^1\lambda(t)g_{\gamma}(t)dt$$ for some $\lambda\geq 0$ and $\beta< 1$ and if, in addition $\Pi_{\gamma}(t)/(1-t^2)$ is decreasing on (0,1), the authors prove their principal result, that states that $V_{\lambda}(\mathcal {P}_{\lambda}(\beta))\subset S^*$, where $S^*$ is the subclass of $\mathcal A$ consisting of starlike functions in $\Delta$. Some other results on the class $\mathcal P_{\gamma}(\beta)$ and applications are also given.
[Eugen Drăghici (Sibiu)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions

Keywords: Hadamard product; univalent function; starlike function

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