manuscripta mathematica

, Volume 112, Issue 4, pp 519–538 | Cite as

Length functions of lemniscates

  • O.S. Kuznetsova
  • V.G. TkachevEmail author


We study metric and analytic properties of generalized lemniscates E t (f)={z∈ℂ:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E t (f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E t (f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulae of the length function in some special cases.


Analytic Function Result State Explicit Formula Analytic Property Positive Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis, English translation. Oliver and Boyd, Edingburgh, 1965Google Scholar
  2. 2.
    Alzer, H., Berg, C.: Some classes of completely monotonic functions. Ann. Acad. Sci. Fen. Math. 27, 445–460 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. New-York-Berlin, Springer-Verlag. 1984Google Scholar
  4. 4.
    Berg, C., Duran, A.J.: A transformation from Hausdorff to Stieltjes moment sequences (to appear)Google Scholar
  5. 5.
    Bernstein, S.N.: Sur les fonctions absolument monotones. Acta math 52, 1–66 (1928)zbMATHGoogle Scholar
  6. 6.
    Borwein, P.: The arc length of the lemniscate |P(z)|=1. Proc. Am. Math. Soc. 123, 797–799 (1995)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Butler, J.P.: The perimeter of a rose. Am. Math. Monthly. 98 (2), 139–143 (1991)zbMATHGoogle Scholar
  8. 8.
    Duren, P.L.: Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York 1983Google Scholar
  9. 9.
    Elia, M., Galizia Angeli, M.T.: The length of a lemniscate. Publ. Inst. Math. Beograd (N.S.) 36, 51–55 (1984)Google Scholar
  10. 10.
    Erdelyi, A. (ed.): Higher transcendental functions. Vol. I. Bateman Manuscript Project, California Institute of Technology. Malabar, Florida: Robert E. Krieger Publishing Company. XXVI, 1981Google Scholar
  11. 11.
    Erdélyi, T.: Paul Erdös and polynomials. Jour. of Appr. Theory, 94, 2–14 (1998)Google Scholar
  12. 12.
    Erdös, P.: Some old and new problems in approximation theory: research problem 95-1. Constr. Approx. 11, 419–421 (1995)Google Scholar
  13. 13.
    Erdös, P., Herzog, F., Piranian, G.: Metric properties of polynomials. J. D’Analyse Math. 6, 125–148 (1958)Google Scholar
  14. 14.
    Eremenko, A., Hayman, W.: On the length of lemniscates. Mich. Math. J. 46, 409–415 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Federer, H.: Geometric Measure Theory. Classics in Mathematics. Berlin: Springer-Verlag. xvi, 1996Google Scholar
  16. 16.
    Giordano, C., Palumbo, B., Pečari, J.: Remarks on the Hankel determinants inequalities. Rend. Circ. Mat. Di Palermo, Serie II XLVI, 279–286 (1997)Google Scholar
  17. 17.
    Hamburger, H.: Über eine Erweiterung des Stieltjesschen Momentenproblems. Math. Ann. 81, (1920)Google Scholar
  18. 18.
    Hille, E.: Analytic function theory. Vol. II. Ginn & Co. New York, 1962Google Scholar
  19. 19.
    Kimberling, C.H.: A probabilistic interpretation of complete monotonicity. Aequations Math. 10, 152–164 (1974)zbMATHGoogle Scholar
  20. 20.
    Klyachin, V.A.: New examples of tubular minimal surfaces of arbitrary codimension. Math. Notes 62 (1-2), 129–131 (1998)Google Scholar
  21. 21.
    Kuznetsova, O.S., Tkachev, V.G.: Analysis on lemniscates and Hamburger’s moments. Preprint. TRITA-MAT-2003-04. Division of Math., Royal Inst. of Techn., Stockholm, 2003Google Scholar
  22. 22.
    Marden, M.: The Geometry of Zeros of a Polynomial in a Complex Variable. Vol. 3 of Mathematics Surveys. Amer. Math. Soc., Providence, 1949Google Scholar
  23. 23.
    Mikljukov, V.M.: Some properties of tubular minimal surfaces in R n. Dokl. Akad. Nauk SSSR 247 (3), 549–552 (1979)Google Scholar
  24. 24.
    Miklyukov, V.M., Tkachev, V.G.: Some properties of tubular minimal surfaces of arbitrary codimension. Math. USSR, Sb. 68 (1), 133–150 (1990)Google Scholar
  25. 25.
    Miller, K.S, Samko, S.G.: Completely monotonic functions. Integr. transf. and special funct. 12 (4), 389–402 (2001)zbMATHGoogle Scholar
  26. 26.
    Piranian, G.: The length of a lemniscate. Am. Math. Month. 87, 555–556 (1980)Google Scholar
  27. 27.
    Pommerenke, Ch.: On some problems of Erdös, Herzog and Piranian. Mich. Math. J. 6, 221–225 (1959)CrossRefzbMATHGoogle Scholar
  28. 28.
    Pommerenke, Ch.: On some metric properties of polynomials II. Mich. Math. J. 8, 49–54 (1961)CrossRefzbMATHGoogle Scholar
  29. 29.
    Umemura, Y.: Measures on infinite dimensional vector spaces. Publ RIMS. Kyoto Univ. 1, 1–47 (1965)Google Scholar
  30. 30.
    Widder D.V.: The Laplace Transform. Princeton, University Press, Princeton, 1946Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Royal Institute of TechnologyStockholmSweden
  2. 2.Volgograd State UniversityVolgogradRussia

Personalised recommendations