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Zbl 1130.30307
Kanas, Stanisława
Techniques of the differential subordination for domains bounded by conic sections.
(English)
[J] Int. J. Math. Math. Sci. 2003, No. 38, 2389-2400 (2003). ISSN 0161-1712; ISSN 1687-0425/e

Summary: We solve the problem of finding the largest domain $D$ for which, under given $\psi$ and $q$, the differential subordination $\psi(p(z), zp^{\prime}(z)) \in D \Rightarrow p(z) \prec q(z)$, where $D$ and $q(\cal{U})$ are regions bounded by conic sections, is satisfied. The shape of the domain $D$ is described by the shape of $q(\cal{U})$. Also, we find the best dominant of the differential subordination $p(z) +({zp^{\prime}(z)}/({\beta p(z) + \gamma})) \prec p_{k}(z)$, when the function $p_k$ $(k\in [0,\infty))$ maps the unit disk onto a conical domain contained in a right half-plane. Various applications in the theory of univalent functions are also given.
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
34A25 Analytical theory of ODE
33E05 Elliptic functions and integrals
30C35 General theory of conformal mappings

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