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Complex Polynomials and Maximal Ranges: Background and Applications

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Recent Progress in Inequalities

Part of the book series: Mathematics and Its Applications ((MAIA,volume 430))

Abstract

This survey is dedicated to the discussion of the various aspects of the notion of maximal polynomial ranges. These are the unions of ranges of polynomials restricted by a geometrical condition. The theory of maximal ranges in essentially constructive and permits in many cases the identification of extremal functions. It thereby leads to a unified approach to many old and new inequalities for polynomials. We also discuss the relation of this concept to the approximation of conformai maps in the unit disk by univalent polynomials.

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References

  1. L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Princeton, N.J., 1966.

    MATH  Google Scholar 

  2. V. Andrievskii Approximation of harmonic functions in compact sets in ℂ, Ukrain. Mat. Zh. 45 (1993), 1467–1475.

    Article  MathSciNet  Google Scholar 

  3. V. Andrievskii and St. Ruscheweyh Maximal polynomial subordination to univalent functions in the unit disk, Constr. Approx. 10 (1994), 131–144.

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Andrievskii and St. Ruscheweyh, The maximal range problem for a bounded domain (to appear).

    Google Scholar 

  5. A. Córdova and St. Ruscheweyh On maximal ranges of polynomial spaces in the unit disk, Constr. Approx. 5 (1989), 309–328.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Córdova and St. Ruscheweyh On maximal polynomial ranges in circular domains, Complex Variables Theory Appl. 10 (1988), 295–309.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Córdova and St. Ruscheweyh On the maximal range problèm for slit domains, Proc. of a Conference, Valparaiso, 1989, Lect. Notes Math. 1435, Springer, Berlin — Heidelberg — New York, 1990, pp. 33-44.

    Google Scholar 

  8. A. Córdova and St. Ruscheweyh Subordination of polynomials, Rocky Mountain J. Math. 21 (1991), 159–170.

    Article  MathSciNet  MATH  Google Scholar 

  9. ——, On the univalence of the extremal polynomials in the maximal range problèm (to appear).

    Google Scholar 

  10. V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials, Nauka, Moscow, 1977. (Russian)

    Google Scholar 

  11. E. Egerváry Abbildungseigenschaften der arithmetischen Mittel der geometrischen Reihe, Math. Z. 42 (1937), 221–230.

    Article  MathSciNet  Google Scholar 

  12. F. W. Gehring, Characteristic Properties of Quasidisks, Les Presses de l’Université de Montréal, 1982.

    Google Scholar 

  13. R. Greiner, Zur Güte der Approximation Schlichter Abbildungen durch Maximal Subordinierende Polynomfolgen, Diplomarbeit, Würzburg, 1993.

    Google Scholar 

  14. R. Greiner and St. Ruscheweyh, On the approximation of univalent functions by subordinate polynomials in the unit disk, Approximation and Computation (R.V.M. Zahar, ed.), ISNM 119, Birkhäuser, Boston, 1994, pp. 261-271.

    Google Scholar 

  15. C. Günther, St. Ruscheweyh, and L. Salinas, New maximal ranges (to appear).

    Google Scholar 

  16. F. Holland Some extremum problems for polynomials with positive real part, Bull. London Math. Soc. 5 (1973), 54–58.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Lachance, E. B. Saff and R. S. Varga Inequalities for polynomials with a prescribed zero, Math. Z. 168 (1979), 105–116.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Lavrentiev Sur la continuité des fonctions univalentes, C.R. Acad. Sci. USSR 4 (1936), 215–217.

    Google Scholar 

  19. J. Lewis Applications of a convolution theorem to Jacobi polynomials, SIAM J. Math. Anal. 10 (1979), 1110–1120.

    Article  MathSciNet  MATH  Google Scholar 

  20. O. Martio and J. Sarvas Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn., Ser. A I Math. 4 (1978), 383–401.

    MathSciNet  Google Scholar 

  21. G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer, New York, 1967.

    Book  MATH  Google Scholar 

  22. H. P. Mulholland On two extremal problems for polynomials in the unit circle, J. London Math. Soc. 31 (1956), 191–199.

    Article  MathSciNet  MATH  Google Scholar 

  23. Chr. Pommerenke, Boundary Bevaviour of Conformai Maps, Springer, New York, 1992.

    Book  Google Scholar 

  24. St. Ruscheweyh Geometric properties of Cesàro means, Resultate Math. 22 (1992), 739–748.

    MathSciNet  MATH  Google Scholar 

  25. St. Ruscheweyh and R. S. Varga On the minimum moduli of normalized polynomials with two prescribed values, Constr. Approx. 2 (1986), 349–369.

    Article  MathSciNet  MATH  Google Scholar 

  26. St. Ruscheweyh and K.J. Wirths On an extremal problem for bounded polynomials, Approx. Theory Appl. 1 (1985), 115–125.

    MathSciNet  MATH  Google Scholar 

  27. S. Smale The fundamental theorem of algebra and complexity theory, Bull. Amer. Math. Soc. 4 (1981), 1–36.

    Article  MathSciNet  MATH  Google Scholar 

  28. T. J. Suifridge On univalent polynomials, J. London Math. Soc. 44 (1969), 496–504.

    Article  MathSciNet  Google Scholar 

  29. T. J. Suifridge, Starlike functions as limits of polynomials, Advances in Complex Function Theory (Maryland, 1973/1974), Lect. Notes Math. 505, Springer, Berlin — Heidelberg — New York, 1976, pp. 164-203.

    Google Scholar 

  30. G. Szegö On conjugate trigonometric polynomials, Amer. J. Math. 65 (1943), 532–536.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Applied Mathematics and Mechanics of the National Ukrainian Academy of Sciences, Rozy Luxemburg 74, Donetsk, 340114, Ukraine

    Vladimir V. Andrievskii

  2. Mathematisches Institut, Universität Würzburg, D-97074, Würzburg, Germany

    Stephan Ruscheweyh

Authors
  1. Vladimir V. Andrievskii
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  2. Stephan Ruscheweyh
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Editor information

Editors and Affiliations

  1. University of Niš, Niš, Yugoslavia

    G. V. Milovanović (Faculty of Electronic Engineering) (Faculty of Electronic Engineering)

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Andrievskii, V.V., Ruscheweyh, S. (1998). Complex Polynomials and Maximal Ranges: Background and Applications. In: Milovanović, G.V. (eds) Recent Progress in Inequalities. Mathematics and Its Applications, vol 430. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9086-0_3

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  • DOI: https://doi.org/10.1007/978-94-015-9086-0_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4945-2

  • Online ISBN: 978-94-015-9086-0

  • eBook Packages: Springer Book Archive

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