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Transactions of the American Mathematical Society

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Gelfer functions, integral means, bounded mean oscillation, and univalency


Author: Shinji Yamashita
Journal: Trans. Amer. Math. Soc. 321 (1990), 245-259
MSC: Primary 30C45; Secondary 30C50
DOI: https://doi.org/10.1090/S0002-9947-1990-1010891-X
MathSciNet review: 1010891
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Abstract: A Gelfer function $ f$ is a holomorphic function in $ D = \{ \left\vert z \right\vert < 1\} $ such that $ f(0) = 1$ and $ f(z) \ne - f(w)$ for all $ z$, $ w$ in $ D$. The family $ G$ of Gelfer functions contains the family $ P$ of holomorphic functions $ f$ in $ D$ with $ f(0) = 1$ and Re $ f > 0$ in $ D$. If $ f$ is holomorphic in $ D$ and if the $ {L^2}$ mean of $ f'$ on the circle $ \{ \left\vert z \right\vert = r\} $ is dominated by that of a function of $ G$ as $ r \to 1 - 0$, then $ f \in BMOA$. This has two recent and seemingly different results as corollaries. A core of the proof is the fact that $ {\operatorname{log}}f \in BMOA$ if $ f \in G$. Besides the properties obtained concerning $ f \in G$ itself, we shall investigate some families of functions where the roles played by $ P$ in Univalent Function Theory are replaced by those of $ G$. Some exact estimates are obtained.


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  • [1] A. Baernstein II, Univalence and bounded mean oscillation, Michigan Math. J. 23(1976), 217-223. MR 0444935 (56:3281)
  • [2] J. Becker, Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 225(1972), 23-43. MR 0299780 (45:8828)
  • [3] C. Bennett and M. Stoll, Derivatives of analytic functions and bounded mean oscillation, Arch. Math. 47(1986), 438-442. MR 870281 (88a:30074)
  • [4] J. E. Brown, Derivatives of close-to-convex functions, integral means and bounded mean oscillation, Math. Z. 178(1981), 353-358. MR 635204 (82j:30046)
  • [5] J. A. Cima and K. E. Petersen, Some analytic functions whose boundary values have bounded mean oscillation, Math. Z. 147(1976), 237-247. MR 0404631 (53:8431)
  • [6] J. A. Cima and G. Schober, Analytic functions with bounded mean oscillation and logarithms of $ {H^p}$ functions, Math. Z. 151(1976), 295-300. MR 0425128 (54:13085)
  • [7] N. Danikas, Über die BMOA-Norm von $ {\operatorname{log}}(1 - z)$, Arch. Math. 42(1984), 74-75. MR 751474 (86f:42015)
  • [8] P. L. Duren, Theory of $ {H^p}$ spaces, Pure and Appl. Math., vol. 38, Academic Press, New York-San Francisco-London, 1970. MR 0268655 (42:3552)
  • [9] -, Univalent functions, Grundlehren Math. Wiss., vol. 259, Springer-Verlag, New York-Berlin-Heidelberg-Tokyo, 1983. MR 708494 (85j:30034)
  • [10] S. A. Gelfer, (C. A. [ill]), [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] functions, assuming no pair of values $ w$ and $ - w$.)
  • [11] D. Girela, Integral means and BMOA-norms of logarithms of univalent functions, J. London Math. Soc. (2) 33(1986), 117-132. MR 829393 (87k:30026)
  • [12] -, BMO, $ {A_2}$-weights and univalent functions, Analysis 7(1987), 129-143. MR 885120 (88e:30050)
  • [13] A. W. Goodman, Univalent functions. I, II, Mariner, Tampa, Florida, 1983.
  • [14] A. Z. Grinshpan (A. [ill]), [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] [ill] transl.: On the coefficients of univalent functions assuming no pair of values $ w$ and $ - w$, Math. Notes 11(1972), 3-11.
  • [15] J. A. Hummel, A variational method for Gelfer functions, J. Analyse Math. 30(1976), 271-280. MR 0440025 (55:12906)
  • [16] J. A. Jenkins, On Bieberbach-Eilenberg functions. Trans. Amer. Math. Soc. 76(1954), 389-396. MR 0062831 (16:24e)
  • [17] A. J. Lohwater, G. Piranian, and W. Rudin, The derivative of a schlicht function, Math. Scand, 3(1955), 103-106. MR 0072218 (17:249b)
  • [18] T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc. 104(1962), 532-537. MR 0140674 (25:4090)
  • [19] S. Yamashita, Almost locally univalent functions, Monatsh. Math. 81(1976), 235-240. MR 0407263 (53:11042)
  • [20] -, Schlicht holomorphic functions and the Riccati differential equation, Math. Z. 157(1977), 19-22. MR 0486487 (58:6216)
  • [21] -, F. Riesz's decomposition of a subharmonic function, applied to BMOA, Boll. Un. Mat. Ital. (6) 3-A(1984), 103-109. MR 739196 (85g:31001)
  • [22] -, A gap series with growth conditions and its applications, Math. Scand. 60(1987), 9-18. MR 908825 (88m:30002)

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DOI: https://doi.org/10.1090/S0002-9947-1990-1010891-X
Article copyright: © Copyright 1990 American Mathematical Society

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