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A criterion associated with the schlicht Bloch constant. (English) Zbl 0764.30018
Some years ago the author discovered a criterion for an extremal domain for the schlicht Bloch constant which eliminates many examples used in estimating it from above from providing extremal functions, for example, that given by R. E. Goodman [Bull. Am. Math. Soc. 51, 234-239 (1945)]. Not long ago the author mentioned this criterion to Hummel and an unsatisfactory vague version of it is quoted in the paper by E. Beller and J. A. Hummel [Complex Variables, Theory Appl. 4, 243- 252 (1985; Zbl 0568.30008)]. This paper provides a complete account.
Reviewer: J.A.Jenkins

MSC:
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30D45 Normal functions of one complex variable, normal families
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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[1] E. BELLER AND J. HUMMEL, On the univalent Bloch constant, Complex Variables, vol. 4, 1985, pp. 243-252. · Zbl 0568.30008
[2] G. M. GOLUSIN, Interior problems of the theory of univalent functions, Uspekh Mathematicheskii Nauk, vol. 6, 1939, pp. 26-89, (Russian) (Translated under auspices of Office of Naval Research 1947.) Zentralblatt MATH: · Zbl 0063.01673 · www.zentralblatt-math.org
[3] JAMES A. JENKINS, Univalent Functions and Conformal Mapping, Springer Verlag, Berlin-Gottingen-Heidelberg, 1958 · Zbl 0083.29606
[4] M. A. LAVRENTIEV, On the theory of conformal mappings, Trudy Matematicheskogo Instituta imeni V. A. Steklova, No. 5, 1934, pp. 159-245, (Russian) (Translated in A. M. S. Translations, vol. 122, 1984, pp. 1-63.)
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