## Integral transforms of a class of analytic functions.(English)Zbl 1146.30009

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.

### MSC:

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

### Keywords:

Hadamard product; univalent function; starlike function
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### References:

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