## Univalency of weighted integral transforms of certain functions.(English)Zbl 1098.30016

Let $$\mathcal A$$ be the class of analytic functions in the unit disc $$\mathcal U$$ normalized by $$f(0)=f'(0)-1=0.$$ For $$\beta<1$$ and $$\gamma \geq0,$$ let $$f\in \mathcal P_\gamma(\beta)\subset \mathcal A$$ if and only if $Re \left\{ e^{i\eta} \left( (1-\gamma) \frac{f(z)}{z}+ \gamma f'(z)-\beta\right)\right\}>0, \quad z\in\mathcal U,$ for some real number $$\eta.$$ Also for $$f\in\mathcal A$$ let define the integral transform $V_\lambda(f)(z)=\int_0^1 \lambda(t)\frac{f(tz)}{t}dt.$ In this paper the authors give sharp conditions when (i) $$f\in {\mathcal P}_\gamma(\beta),$$ $$\gamma\in[1,\infty)$$ $$\Rightarrow$$ $$V_\lambda(f)\in {\mathcal P}_1(\beta');$$ (ii) $$f\in {\mathcal P}_0(\beta)$$ $$\Rightarrow$$ $$V_\lambda(f)\in {\mathcal P}_1(\beta').$$ The second implication is studied only for some special choices of $$\lambda(t).$$

### MSC:

 30C55 General theory of univalent and multivalent functions of one complex variable 33C55 Spherical harmonics
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### References:

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