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
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
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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
Córdova A. and Ruscheweyh St., On Maximal Ranges of Polynomial Spaces in the Unit Disk, Constructive Approximation 5 (1989), 309–327.
Córdova A. and Ruscheweyh St., On Maximal Polynomials Ranges on Circular Domains, Complex Variables 10 (1988), 295–309.
Córdova A. and Ruscheweyh St., On the Univalence of Extremal Polynomials for the Maximal Range Problem, to appear.
Suffridge, T.J., On Univalent Polynomials, J. London Math. Soc. 44 (1969), 496–504.
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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