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Univalent α-Spiral Functions

  • Richard J. Libera (a1)
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Suppose ƒ is regular in the open unit disk Δ, |z| < 1, and has a simple zero at the origin and no other zeros. Špaček (15) essentially showed that ƒ is univalent in Δ if and only if

such that 0 < r < 1 and 0 < t2 t1 ⩽ 2π.

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References
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1. Clunie, J., On meromorphic schlicht functions, J. London Math. Soc., 34 (1959), 215216.
2. Epstein, B. and Schoenberg, I. J., On a conjecture concerning schlicht functions, Bull. Amer. Math. Soc., 65 (1959), 273275.
3. Kaplan, W., Close-to-convex schlicht functions, Michigan Math. J., 1 (1952), 169185.
4. Krzyź, J., The radius of close-to-convexity within the family of univalent functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 10, 4 (1962), 201204.
5. Krzyź, J. and Lewandowski, Z., On the integral of univalent functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 11, 7 (1963), 447448.
6. Libera, R. J., Some radius of convexity problems, Duke Math. J., 31 (1964), 143158.
7. Marx, A., Untersuchungen iiber schlichte Abbildungen, Math. Ann., 107 (1932), 4067.
8. Montel, P., Leçons sur les fonctions univalentes ou multivalentes (Paris, 1933).
9. Nehari, Z., Conformai mapping (New York, 1952).
10. Ozaki, S., Some remarks on the univalency and multivalency of functions, Sci. Rep. Tokyo Bunrika Daigaku. Sect. A, 2, 32 (1934), 1529.
11. Robertson, M. S., On the theory of univalent functions, Ann. of Math., 37 (1936), 374408.
12. Robertson, M. S., Variational methods for functions with positive real part, Trans. Amer. Math. Soc., 102 (1962), 8293.
13. Robertson, M. S., Radii of starlikeness and close-to-convexity, Proc Amer. Math. Soc., 16 (1965), 847852.
14. Schild, A., On starlike functions of order α, Amer. J. Math., 87, 1 (1965), 6570.
15. Špaček, L., Přispěvek k teorii funcki prostyčh, Căsopis Pěst. Mat. a Fys., 62 (1932), 1219.
16. Zamorski, J., About the extremal spiral schlicht functions, Ann. Polon. Math., 9 (1962), 265273.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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