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On the distribution of zeros of a strongly annular function

Published online by Cambridge University Press:  22 January 2016

Akio Osada
Akio Osada*
Affiliation:
Gifu College of Pharmacy
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A function f(z), regular in the unit disk D, is called annular ([1], p. 340) if there is a sequence of closed Jordan curves Jn ⊂ D satisfying

  1. (A1) Jn is contained in the interior of Jn+1 for every n,

  2. (A2) given ε > 0, there exists a positive number n(ε) such that, for each n > n(ε), Jn lies in the region 1 – ε < | z | < 1 and

  3. (A3) lim min {| f(z) |; z ∈ Jn} = + ∞.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 56 , January 1975 , pp. 13 - 17
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

[1] Bagemihl, F. and Erdös, P., A problem concerning the zeros of a certain kind of holomorphic function in the unit disk, J. Reine Angew. Math., 214/215 (1964), 340–344.Google Scholar
[2] Bagemihl, F. and Seidel, W., Some boundary properties of analytic functions, Math. Z., 61 (1954), 186–199.Google Scholar
[3] Barth, K. and Schneider, W., On a problem of Bagemihl and Erdös concerning the distribution of zeros of an annular function, J. Reine Angew. Math., 234 (1969), 179–183.Google Scholar
[4] Bonar, D. D., On annular functions, Berlin 1971.Google Scholar
[5] Bonar, D. D. and Carrol, F. W., Distribution of a-points for unbounded analytic functions, J. Reine Angew. Math., 253 (1972), 141–145.Google Scholar
[6] Schneider, W., An entire transcendental function whose invers takes set of finite measure into set of finite measure, Bull. Amer. Math. Soc., 72 (1966), 841–842.Google Scholar