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Coefficients of Functions with Bounded Boundary Rotation

  • M. S. Robertson (a1)

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For fixed k ≧ 2, let Vk denote the class of normalized analytic functions

such that zE = {z; |z| <1} are regular and have f′(0) = l,f′(z) ≠ 0, and

1

Let Sk be the subclass of Vk whose members f(z) are univalent in E. It was pointed out by Paatero (4) that Vk coincides with Sk whenever 2 ≦ k ≦ 4. Later Rényi (5) showed that in this case, f(z) ∈ Sk is also convex in one direction in E. In (6) I showed that the Bieberbach conjecture

holds for functions convex in one direction.

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References

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1. Golusin, G. M., On distortion theorems and coefficients of univalent functions, Mat. Sb. 19 (1946), 183202.
2. Hayman, W. K., On successive coefficients of univalent functions, J. London Math. Soc. 88 (1963), 228243.
3. Lehto, O., On the distortion of conformai mappings with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. Al Math. Phys. 124 (1952), 14pp.
4. Paatero, V., Über die konforme Abbildung von Gebieten deren Rdnder von beschrankter Drehung sind, Ann. Acad. Sci. Fenn. Ser. A (83) 9 (1931), 77pp.
5. Rényi, A., On the coefficients of schlicht functions, Publ. Math. Debrecen 1 (1949), 1823.
6. Robertson, M. S., Analytic functions starlike in one direction, Amer. J. Math. 58 (1936), 465472.
7. Robertson, M. S., A generalization of the Bieberbach coefficient problem for univalent functions, Michigan Math. J. 18 (1966), 185192.
8. Schiffer, M. and Tammi, O., On the fourth coefficient of univalent functions with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. Al 896 (1967), 26pp.
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