Annales Academię Scientiarum Fennicę
Mathematica
Volumen 34, 2009, 353-378
Far Eastern Branch of Russian Academy of Sciences, Institute of Applied Mathematics
7 Radio Street, Vladivostok, 690041, Russia; dmkrp@yandex.ru
Far Eastern Branch of Russian Academy of Sciences, Institute of Applied Mathematics
7 Radio Street, Vladivostok, 690041, Russia; pril-elena@yandex.ru
Abstract. We derive an asymptotic formula for the modulus (= reciprocal of capacity) of generalized condenser whose field is an arbitrary multiply-connected domain on the complex sphere and whose plates degenerate into a finite number of inner and/or boundary points of the field. We call the constant term in this asymptotic formula the reduced modulus with free boundary. Our modulus generalizes several previously introduced concepts. The asymptotic formula is given in terms of a generalized version of the classical Neumann function. This generalized Neumann function is introduced in the paper and its properties are studied. The usefulness of the new modulus is illustrated by two applications: a two-point distortion theorem for univalent functions defined in annulus and preserving the unit circle and an inequality for the quadratic form in the difference of the Neumann and Robin functions.
2000 Mathematics Subject Classification: Primary 31A15, 30C85, 30C99.
Key words: Capacity, condenser, Dirichlet principle, reduced modulus, Neumann function, distortion theorem.
Reference to this article: D. Karp and E. Prilepkina: Reduced modulus with free boundary and its applications. Ann. Acad. Sci. Fenn. Math. 34 (2009), 353-378.
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