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Polynomials Versus Finite Blaschke Products

  • Tuen Wai NgEmail author
  • Chiu Yin Tsang
Chapter
Part of the Fields Institute Communications book series (FIC, volume 65)

Abstract

The aim of this chapter is to compare polynomials of one complex variable and finite Blaschke products and demonstrate that they share many similar properties. In fact, we collect many known results as well as some very recent results for finite Blaschke products here to establish a dictionary between polynomials and finite Blaschke products.

Keywords

Polynomials Finite Blaschke products Ritt’s theorems Chebyshev polynomials Approximation 

Mathematics Subject Classification

30J10 30C10 30E10 30D05 39B12 

Notes

Acknowledgements

The first author was partially supported by RGC grant HKU 704409P. The second author was partially supported by graduate studentship of HKU and RGC grant HKU 704409P.

References

  1. 1.
    Adams, W.W., Straus, E.G.: Non-Archimedean analytic functions taking the same values at the same points. Ill. J. Math. 15, 418–424 (1971) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Akhiezer, N.I.: Elements of the Theory of Elliptic Functions. AMS, Providence (1990) zbMATHGoogle Scholar
  3. 3.
    Beardon, A.F.: Iteration of Rational Functions. Springer, Berlin (1991) zbMATHCrossRefGoogle Scholar
  4. 4.
    Beardon, A.F., Carne, T.K., Ng, T.W.: The critical values of a polynomial. Constr. Approx. 18, 343–354 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Beardon, A.F., Minda, D.: A multi-point Schwarz-Pick lemma. J. Anal. Math. 92, 81–104 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Beardon, A.F., Minda, D., Ng, T.W.: Smale’s mean value conjecture and the hyperbolic metric. Math. Ann. 322, 623–632 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics, vol. 161. Springer, New York (1995) zbMATHCrossRefGoogle Scholar
  8. 8.
    Carleson, L., Gamelin, T.W.: Complex Dynamics. Springer, Berlin (1993) zbMATHCrossRefGoogle Scholar
  9. 9.
    Chandrasekharan, K.: Elliptic Functions. Springer, Berlin (1985) zbMATHGoogle Scholar
  10. 10.
    Conte, A., Fujikawa, E., Lakic, N.: Smale’s mean value conjecture and the coefficients of univalent functions. Proc. Am. Math. Soc. 135, 3295–3300 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Crane, E.: A bound for Smale’s mean value conjecture for complex polynomials. Bull. Lond. Math. Soc. 39, 781–791 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dinh, T.C.: Ensembles d’unicité pour les polynômes. Ergod. Theory Dyn. Syst. 22(1), 171–186 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Dinh, T.C.: Distribution des préimages et des points périodiques d’une correspondance polynomiale. Bull. Soc. Math. Fr. 133, 363–394 (2005) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Farkas, H.M., Kra, I.: Theta Constants, Riemann Surfaces and the Modular Group. Grad. Stud. Math., vol. 37. Am. Math. Soc., Providence (2001) zbMATHGoogle Scholar
  15. 15.
    Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 47, 161–271 (1919) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 48, 33–94 (1920) MathSciNetGoogle Scholar
  17. 17.
    Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. Fr. 48, 208–314 (1920) MathSciNetGoogle Scholar
  18. 18.
    Fatou, P.: Sur les fonctions holomorphes et bornées à l’intérieur d’uncercle. Bull. Soc. Math. Fr. 51, 191–202 (1923) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Forster, O.: Lectures on Riemann Surfaces. Springer, New York (1991) Google Scholar
  20. 20.
    Fujikawa, E., Sugawa, T.: Geometric function theory and Smale’s mean value conjecture. Proc. Jpn. Acad., Ser. A, Math. Sci. 82, 97–100 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Garnett, J.B.: Bounded Analytic Functions. Grad. Texts in Math., vol. 236. Springer, New York (2007) Google Scholar
  22. 22.
    Gonchar, A.A.: Zolotarev problems connected with rational functions. Math. USSR Sb. 7, 623–635 (1969) zbMATHCrossRefGoogle Scholar
  23. 23.
    Hinkkanen, A., Kayumov, I.: Smale’s problem for critical points on certain two rays. J. Aust. Math. Soc. 88, 183–191 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Horwitz, A.L., Rubel, L.A.: A uniqueness theorem for monic Blaschke products. Proc. Am. Math. Soc. 96, 180–182 (1986) MathSciNetzbMATHGoogle Scholar
  25. 25.
    Istace, M.P., Thiran, J.P.: On the third and fourth Zolotarev problems in the complex plane. SIAM J. Numer. Anal. 32, 249–259 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lorentz, G.G.: Approximation of Functions. Chelsea, New York (1986) zbMATHGoogle Scholar
  27. 27.
    Mason, J.C., Handscomb, D.: Chebyshev Polynomials. Chapman and Hall, London (2003) zbMATHGoogle Scholar
  28. 28.
    Milnor, J.: Dynamics in One Complex Variable. Princeton University Press, Princeton (2006) zbMATHGoogle Scholar
  29. 29.
    Ng, T.W.: Smale’s mean value conjecture for odd polynomials. J. Aust. Math. Soc. 75, 409–411 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Ng, T.W., Tsang, C.Y.: Chebyshev-Blaschke products. Preprint (2012) Google Scholar
  31. 31.
    Ng, T.W., Tsang, C.Y.: On finite Blaschke products sharing preimages of sets. Preprint (2012) Google Scholar
  32. 32.
    Ng, T.W., Wang, M.X.: Ritt’s theory on the unit disk. Forum Math. doi: 10.1515/form.2011.136
  33. 33.
    Ostrovskii, I.V., Pakovich, F.B., Zaidenberg, M.G.: A remark on complex polynomials of least deviation. Int. Math. Res. Not. 14, 699–703 (1996) MathSciNetCrossRefGoogle Scholar
  34. 34.
    Pakovich, F.B.: Sur un problème d’unicité pour les polynômes, prépublication. Inst. Fourier Math. 324, 1–4 (1995). Grenoble Google Scholar
  35. 35.
    Pakovich, F.B.: Sur un problème d’unicité pour les fouctions méromorphes. C.R. Acad. Sci. Paris Sr. I Math. 323, 745–748 (1996) MathSciNetGoogle Scholar
  36. 36.
    Pakovich, F.B.: On polynomials sharing preimages of compact sets and related questions. Geom. Funct. Anal. 18, 163–183 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Pritsker, I.E.: Products of polynomials in uniform norms. Trans. Am. Math. Soc. 353, 3971–3993 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Pritsker, I.E.: Inequalities for sums of Green potentials and Blaschke products. Bull. Lond. Math. Soc. 43, 561–575 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Radó, T.: Zur Theorie der mehrdeutigen konformen Abbildungen. Acta Litt. Sci. Univ. Hung. 1, 55–64 (1922) (German) zbMATHGoogle Scholar
  40. 40.
    Remmert, R.: Classical Topics in Complex Function Theory. Springer, New York (1998) zbMATHGoogle Scholar
  41. 41.
    Ritt, J.F.: Prime and composite polynomials. Trans. Am. Math. Soc. 23(1), 51–66 (1922) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Rivlin, T.J.: Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory, 2nd edn. Wiley, New York (1990) zbMATHGoogle Scholar
  43. 43.
    Sheil-Small, T.: Complex Polynomials. Cambridge University Press, Cambridge (2002) zbMATHCrossRefGoogle Scholar
  44. 44.
    Singer, D.A.: The location of critical points of finite Blaschke products. Conform. Geom. Dyn. 10, 117–124 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Smale, S.: The fundamental theorem of algebra and complexity theory. Bull. Am. Math. Soc. 4(1), 1–36 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Steinmetz, N.: Rational Iteration: Complex Analytic Dynamical Systems. Studies in Mathematics (1993) zbMATHCrossRefGoogle Scholar
  47. 47.
    Walker, P.L.: Elliptic Functions: A Constructive Approach. Wiley, New York (1996) zbMATHGoogle Scholar
  48. 48.
    Walsh, J.L.: Note on the location of zeros of extremal polynomials in the non-euclidean plane. Acad. Serbe Sci. Publ. Inst. Math. 4, 157–160 (1952) MathSciNetzbMATHGoogle Scholar
  49. 49.
    Wang, M.X.: Factorizations of finite mappings on Riemann surfaces. M.Phil. thesis, HKU (2008). http://hub.hku.hk/handle/123456789/51854
  50. 50.
    Wang, M.X.: Rational points and transcendental points. Ph.D. thesis, ETH (2011). http://e-collection.library.ethz.ch/view/eth:4704
  51. 51.
    Yang, C.C.: Open problems in complex analysis. In: Proceedings of the S.U.N.Y. Brockport Conference. Dekker, New York (1978) Google Scholar
  52. 52.
    Zakeri, S.: On critical points of proper holomorphic maps on the unit disk. Bull. Lond. Math. Soc. 30, 62–66 (1996) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Hong KongPokfulamHong Kong

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