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Criteria for univalence of certain integral operators. (English) Zbl 1212.30067

Summary: We determine conditions on \(\beta\), \(\alpha _i\) and \(f_i(z)\) so that the integral operator \({ \{\beta \int_0^z \zeta ^{\beta -1} \prod_{i=1}^n {(f_i(\zeta )/\zeta)}^{1/\alpha _i} d\zeta }\}^{1/\beta}\) is univalent in the open unit disk. We also obtain similar results for the integral operator \({\{\beta \int_0^z \zeta ^{\beta -1} {\text{exp}}(\sum_{i=1}^n \alpha _i f_i(\zeta)) d\zeta \}}^{\frac 1\beta}\).

MSC:

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