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Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications

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Abstract

The numerical range of holomorphic mappings arises in many aspects of non-linear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings. In the present paper, we establish lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, non-linear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness.

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References

  1. Aharonov, D., Reich, S., Shoikhet, D.: Flow invariance conditions for holomorphic mappings in Banach spaces. Math. Proc. R. Ir. Acad. 99A, 93–104 (1999)

    MathSciNet  MATH  Google Scholar 

  2. Bracci, F., Elin, M., Shoikhet, D.: Growth estimates for pseudo-dissipative holomorphic maps in Banach spaces. J. Nonlinear Convex Anal. 15, 191–198 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Bracci, F., Kozitsky, Y., Shoikhet, D.: Abel averages and holomorphically pseudo-contractive maps in Banach spaces. J. Math. Anal. Appl. 423, 1580–1593 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chae, S.B.: Holomorphy and Calculus in Normed Spaces. Marcel Dekker, New York (1985)

    MATH  Google Scholar 

  5. Earle, C.J., Hamilton, R.S.: A fixed point theorem for holomorphic mappings. In: Proceedings Symposium Pure Mathematics, Vol. 16, American Mathematical Society, Providence, RI, pp. 61–65 (1970)

  6. Elin, M., Reich, S., Shoikhet, D.: Holomorphically accretive mappings and spiral-shaped functions of proper contractions. Nonlinear Anal. Forum 5, 149–161 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Franzoni, T., Vesentini, E.: Holomorphic Maps and Invariant Distances. North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  8. Goluzin, G.: Geometric Function Theory of Complex Variable. GITTL, Moscow (1952)

    MATH  Google Scholar 

  9. Graham, I., Hamada, H., Kohr, G.: Radius problems for holomorphic mappings on the unit ball in \(\mathbb{C}^{n}\). Math. Nachr. 279, 1474–1490 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Graham, I., Hamada, H., Honda, T., Kohr, G., Shon, K.H.: Distortion and coefficient bounds for Carathéodory families in \(\mathbb{C}^{n}\) and complex Banach spaces. J. Math. Anal. Appl. 416, 449–469 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Harris, L.A.: The numerical range of holomorphic functions in Banach spaces. Am. J. Math. 93, 1005–1019 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  12. Harris, L.A., Reich, S., Shoikhet, D.: Dissipative holomorphic functions, Bloch radii, and the Schwarz lemma. J. Anal. Math. 82, 221–232 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Khatskevich, V., Reich, S., Shoikhet, D.: Global implicit function and fixed point theorems for holomorphic mappings and semigroups. Complex Var. 28, 347–356 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kresin, G., Maz’ya, V.G.: Sharp Real-Part Theorems. A Unified Approach, Lecture Notes in Mathematics, 1903, Springer, Berlin (2007)

  15. Reich, S.: On fixed point theorems obtained from existence theorems for differential equations. J. Math. Anal. Appl. 54, 26–36 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  16. Reich, S., Shoikhet, D.: Generation theory for semigroups of holomorphic mappings in Banach spaces. Abstr. Appl. Anal. 1, 1–44 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reich, S., Shoikhet, D.: Semigroups and generators on convex domains with the hyperbolic metric. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9, 231–250 (1997)

  18. Reich, S., Shoikhet, D.: Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces. Imperial College Press, London (2005)

    Book  MATH  Google Scholar 

  19. Reich, S., Shoikhet, D., Zemánek, J.: Ergodicity, numerical range, and fixed points of holomorphic mappings. J. Anal. Math. 119, 275–303 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Robertson, M.S.: Radii of star-likeness and close-to-convexity. Proc. Am. Math. Soc. 16, 847–852 (1966)

    MATH  Google Scholar 

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Acknowledgments

We are very grateful to the anonymous referees for their many useful comments and helpful suggestions.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Rome, Italy

    Filippo Bracci

  2. Department of Mathematics, ORT Braude College, 21982, Karmiel, Israel

    Marina Levenshtein & David Shoikhet

  3. Department of Mathematics, The Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich

Authors
  1. Filippo Bracci
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  2. Marina Levenshtein
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  3. Simeon Reich
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  4. David Shoikhet
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Corresponding author

Correspondence to Filippo Bracci.

Additional information

Communicated by Stephan Ruscheweyh.

Filippo Bracci was partially supported by the ERC grant “HEVO—Holomorphic Evolution Equations” No. 277691. Simeon Reich was partially supported by the Israel Science Foundation (Grant No. 389/12), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. Marina Levenstein and David Shoikhet were partially supported by the European Commission under the project STREVCOMS PIRSES-2013-612669.

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Bracci, F., Levenshtein, M., Reich, S. et al. Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications. Comput. Methods Funct. Theory 16, 457–487 (2016). https://doi.org/10.1007/s40315-015-0155-6

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  • DOI: https://doi.org/10.1007/s40315-015-0155-6

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