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Schlicht Dirichlet Series

  • M. S. Robertson (a1)

Extract

For power series

(1.1) for which

(1.2) ,

it has been known for four decades (1) that ƒ(z) is regular and univalent or schlicht in |z| < 1. This theorem, due to J. W. Alexander, has more recently been studied by Remak (5) who has shown that w = ƒ(z), under the hypothesis (1.2), maps |z| < 1 onto a star-like region, and if (1.2) is not satisfied=(z) need not be univalent in |z| < 1 for a proper choice of the amplitudes of the coefficients an.

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References

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1. Alexander, J. W., Functions which map the interior of the unit circle upon simple regions, Ann. of Math., 17 (1915), 1222.
2. Goluzin, G. M., On distortion theorems and coefficients of univalent functions, Rec. Math. (Mat. Sbornik), N.S. (61), 19 (1946), 183-202.
3. Montel, P., Sur les fonctions localement univalentes ou multivalentes, Ann. Sci. École Norm. Sup. (3), 54 (1937), 39-54.
4. Noshiro, J., On the theory of schlicht functions, J. Fac. Sci. Hokkaido Univ. (1), 2 (1934), 129155.
5. Remak, R., Ueber eine specielle Klasse schlichter konformer Abbildungen des Einheitskreises, Mathematica, Zutphen, 11 (1943), 175-192; 12 (1943), 43-49.
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Schlicht Dirichlet Series

  • M. S. Robertson (a1)

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