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Univalence conditions of general integral operator. (English) Zbl 1299.30024

Summary: We obtain new univalence conditions for the integral operator \[ I_{\xi}^{\alpha_{i},\beta_i}(f_1,\dots,f_n)(z) =\left[\xi\int_0^zt^{\xi-1}(f'_1(t))^{\alpha_1}(\frac{f_1(t)}t)^{\beta_1}\cdots(f'_n(t))^{\alpha_n} (\frac{f_n(t)}{t})^{\beta_n}\,dt\right]^{\frac{1}{\xi}} \] of analytic functions defined in the open unit disc.

MSC:

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