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Zbl 1098.30016
Barnard, R.W.; Naik, S.; Ponnusamy, S.
Univalency of weighted integral transforms of certain functions.
(English)
[J] J. Comput. Appl. Math. 193, No. 2, 638-651 (2006). ISSN 0377-0427

Let $\cal A$ be the class of analytic functions in the unit disc $\cal U$ normalized by $f(0)=f'(0)-1=0.$ For $\beta<1$ and $\gamma \ge0,$ let $f\in \cal P_\gamma(\beta)\subset \cal A$ if and only if $$Re \left\{ e^{i\eta} \left( (1-\gamma) \frac{f(z)}{z}+ \gamma f'(z)-\beta\right)\right\}>0, \quad z\in\cal U,$$ for some real number $\eta.$ Also for $f\in\cal A$ 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) $f\in {\cal P}_\gamma(\beta),$ $\gamma\in[1,\infty)$ $\Rightarrow$ $V_\lambda(f)\in {\cal P}_1(\beta');$ (ii) $f\in {\cal P}_0(\beta)$ $\Rightarrow$ $V_\lambda(f)\in {\cal P}_1(\beta').$ The second implication is studied only for some special choices of $\lambda(t).$
[Nikola Tuneski (Skopje)]
MSC 2000:
*30C55 General theory of univalent and multivalent functions
33C55 Elliptic integrals as hypergeometric functions

Keywords: subordination; Gausian hypergeometric function; univalent function; starlike function; convex function; Hadamard product

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