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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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