Skip to main content
SpringerLink
Log in

On Zeros of Certain Analytic Functions

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

Given a function s which is analytic and bounded by one in modulus in the open unit disk \({{\mathbb D}}\) and given a finite Blaschke product \({\vartheta}\) of degree k, we relate the number of zeros of the function \({s-\vartheta}\) inside \({{\mathbb D}}\) to the number of boundary zeros of special type of the same function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akhiezer N.I.: On a minimum problem in the theory of functions, and on the number of roots of an algebraic equation which lie inside the unit circle. Izv. Akad. Nauk 9, 1169–1189 (1930)

    Google Scholar 

  2. Ball J.A.: Interpolation problems of Pick-Nevanlinna and Loewner type for meromorphic matrix functions. Integr. Equ. Oper. Theory 6, 804–840 (1983)

    Article  MATH  Google Scholar 

  3. Ball, J.A., Gohberg, I., Rodman, L.: Interpolation of Rational Matrix Functions. OT 45. Birkhäuser (1990)

  4. Ball J.A., Helton J.W.: Interpolation problems of Pick-Nevanlinna and Loewner types for meromorphic matrix functions: parameterization of the set of all solutions. Integr. Equ. Oper. Theory 9, 155–203 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bolotnikov, V.: On a certain generalization of the Carathéodory-Julia-Wolff theorem. Bull. Belg. Math. Soc. Simon Stevin (to appear)

  6. Bolotnikov V.: A uniqueness result on boundary interpolation. Proc. Am. Math. Soc. 136(5), 1705–1715 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bolotnikov V., Dym H.: On degenerate interpolation maximum entropy and extremal problems for matrix Schur functions. Integr. Equ. Oper. Theory 32(4), 367–435 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bolotnikov, V., Dym, H.: On boundary interpolation for matrix Schur functions. Mem. Am. Math. Soc. 181, no 856 (2006)

    Google Scholar 

  9. Bolotnikov, V., Kheifets, A.: The higher order Carathéodory–Julia theorem and related boundary interpolation problems. Oper. Theory Adv. Appl. OT 179, 63–102. Birkhäuser, Basel (2007)

    Google Scholar 

  10. Bolotnikov V., Kheifets A.: Carathéodory-Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk. Math. Nachr. 282(11), 1513–1536 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Julia G.: Extension d’un lemma de Schwartz. Acta Math. 42, 349–355 (1920)

    Article  MathSciNet  Google Scholar 

  12. Katsnelson, V., Kheifets, A., Yuditskii, P.: An abstract interpolation problem and extension theory of isometric operators. Oper. Theory Adv. Appl. OT 95, 283–298. Birkhäuser Verlag, Basel (1997)

    Google Scholar 

  13. Kheifets, A.: Scattering matrices and Parseval equality in abstract interpolation problem. Ph.D. Thesis, Khrakov State University (1990)

  14. Kreĭn M.G., Langer H.: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raume Π κ zusammenhängen. I. Einige Funktionenklassen und ihre Darstellungen. Math. Nachr. 77, 187–236 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  15. Takagi T.: On an algebraic problem related to an analytic theorem of Carathéodory and Fejér. Jpn. J. Math. 1, 83–93 (1924)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Vladimir Bolotnikov

Authors
  1. Vladimir Bolotnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Bolotnikov.

Additional information

V. Bolotnikov was partially supported by National Science Foundation Grant DMS 0901124.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bolotnikov, V. On Zeros of Certain Analytic Functions. Integr. Equ. Oper. Theory 69, 203–215 (2011). https://doi.org/10.1007/s00020-010-1826-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-010-1826-3

Mathematics Subject Classification (2010)

Keywords

Search

Navigation