## 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.)

### Keywords:

starlike function
Full Text:

### References:

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