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A third-order differential equation and starlikeness of a double integral operator. (English) Zbl 1207.30012

Summary: Functions \(f(z)=z+\sum_{n=2}^\infty a_n z^n\) that are analytic in the unit disk and satisfy the differential equation \(f'(z)+\alpha z f''(z)+\beta z^2 f'''(z)=g(z)\) are considered, where \(g\) is subordinated to a normalized convex univalent function \(h\). These functions \(f\) are given by a double integral operator of the form
\[ f(z)=\int_0^1\int_0^1 G(z t^\mu s^\nu) t^{-\mu} s^{-\nu} ds dt \]
with \(G'\) subordinated to \(h\). The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function \(h\).

MSC:

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

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