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Length functions of lemniscates

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Abstract.

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.

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Authors and Affiliations

  1. Royal Institute of Technology, Lindstedtsvägen 25, 10044, Stockholm, Sweden

    O.S. Kuznetsova

  2. Volgograd State University, 2-ya Prodolnaya 30, Volgograd, 400062, Russia

    V.G. Tkachev

Authors
  1. O.S. Kuznetsova
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  2. V.G. Tkachev
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Correspondence to V.G. Tkachev.

Additional information

The author was supported the Göran Gustafsson foundation and grant RFBR no. 03-01-00304.

The author was supported by Russian President grant for young doctorates no. 00-15-99274 and grant RFBR no. 03-01-00304.

Mathematics Subject Classification (2000): 30E05, 42A82, 44A10

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Kuznetsova, O., Tkachev, V. Length functions of lemniscates. manuscripta math. 112, 519–538 (2003). https://doi.org/10.1007/s00229-003-0411-3

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