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On the maximal range problem for slit domains

Part of the Lecture Notes in Mathematics book series (LNM,volume 1435)

Abstract

Let Ω ⊂ ℂ be a domain, 0 ε Ω. For the family P n(Ω) of complex polynomials p of degree ≤ n satisfying p(0) = 0, p(\(\mathbb{D}\)) ⊂ Ω (\(\mathbb{D}\) the unit disk) we define the maximal range Ωn as

$$\Omega _n : = \mathop \cup \limits_{p \in \mathcal{P}_n \left( \Omega \right)} p\left( \mathbb{D} \right).$$

We are interested in the explicit characterization of Ωn for some specific domains as well as the corresponding extremal polynomials p ε P n(ω), i.e. the ones with

. In this paper we solve completely the maximal range problem for the slit domains

These results yield, for instance, new inequalities relating ‖p‖, ¦Rep¦, ¦Imp¦ for typically real polynomials.

Keywords

  • Unit Disk
  • Real Polynomial
  • Extremal Polynomial
  • Explicit Characterization
  • German Academic Exchange

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research supported by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT, Grant 237/89), by the Universidad F. Santa María (Grant 89.12.06), and by the German Academic Exchange Service (DAAD).

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References

  1. Córdova A. and Ruscheweyh St., On Maximal Ranges of Polynomial Spaces in the Unit Disk, Constructive Approximation 5 (1989), 309–327.

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  2. Córdova A. and Ruscheweyh St., On Maximal Polynomials Ranges on Circular Domains, Complex Variables 10 (1988), 295–309.

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  3. Córdova A. and Ruscheweyh St., On the Univalence of Extremal Polynomials for the Maximal Range Problem, to appear.

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  4. Suffridge, T.J., On Univalent Polynomials, J. London Math. Soc. 44 (1969), 496–504.

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  5. Rahman, Q.I. and Ruscheweyh St., Markov's Inequality for Typically Real Polynomials, J. Anal. Appl. (to appear).

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© 1990 Springer-Verlag

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Córdova Yévenes, A., Ruscheweyh, S. (1990). On the maximal range problem for slit domains. In: Ruscheweyh, S., Saff, E.B., Salinas, L.C., Varga, R.S. (eds) Computational Methods and Function Theory. Lecture Notes in Mathematics, vol 1435. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0087895

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  • DOI: https://doi.org/10.1007/BFb0087895

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52768-8

  • Online ISBN: 978-3-540-47139-4

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