Abstract
This paper is a continuation of our earlier work and focuses on the structural and geometric properties of functions in analytic Besov spaces, primarily on univalent functions in such spaces and their image domains. We improve several earlier results.
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P. Ahern, The mean modulus of the derivative of an inner function, Indiana Univ. Math. J. 28 (1979), 311–347.
P. Ahern and D. Clark, On inner functions with Hp derivative, Michigan Math. J. 21 (1974), 115–127.
J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37.
J. Arazy, S. D. Fisher, and J. Peetre, Möbius invariant function spaces, J. Reine Angew.Math. 363 (1985), 110–145.
A. Baernstein, Analytic functions of Bounded Mean Oscillation, in Aspects of Contemporary Complex Analysis (D. A. Brannan and J. G. Clunie, eds.), Academic Press, London, 1980, pp. 3–36.
A. Baernstein, D. Girela and J. A. Peláez, Univalent functions, Hardy spaces and spaces of Dirichlet type, Illinois J. Math. 48 (2004), 837–859.
B. Böe, Interpolating sequences for Besov spaces, J. Funct. Anal. 192 (2002), 319–341.
S.M. Buckley, J. L. Fernández, and D. Vukotič, Superposition operators on Dirichlet type spaces in Papers on Analysis, Rep. Univ. Jyväskylä 83 (2001), 41–61.
S. M. Buckley and D. Vukotić, Univalent interpolation in Besov spaces and superpositions into Bergman spaces, Potential Anal. 29 (2008), 1–16.
V. F. Cowling, A remark on bounded functions, Amer. Math. Monthly 66 (1959), 119–120.
J. J. Donaire, D. Girela, and D. Vukotić, On univalent functions in some Möbius invariant spaces, J. Reine Angew. Math. 553 (2002), 43–72.
P. L. Duren, Univalent Functions, Springer-Verlag, Berlin, 1983.
P. L. Duren, Theory of Hp Spaces, Academic Press, New York-London 1970. Reprint: Dover, Mineola, New York, 2000.
L. Fejér and F. Riesz, Über einige funktionentheoretische Ungleichungen, Math. Z. 11 (1921), 305–314.
W. H. J. Fuchs, On the zeros of power series with Hadamard gaps, Nagoya Math. J. 29 (1967), 167–174.
J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge University Press, Cambridge, 2005.
D. Girela, Analytic Functions of Bounded Mean Oscillation, in Complex Function Spaces, Mekrijärvi 1997 (R. Aulaskari, ed.), Univ. Joensuu Dept. Math. Rep. Ser. 4, Univ. Joensuu, Joensuu, 2001, pp. 61–170.
D. Girela, J.A. Peláez and D. Vukotić, Integrability of the derivative of a Blaschke product, Proc. Edinburgh Math. Soc. 50 (2007), 673–687.
W. K. Hayman, Multivalent Functions, second edition, Cambridge University Press, Cambridge, 1994.
F. Holland and D. Walsh, Growth estimates for functions in the Besov spaces A p, Proc. Roy. Irish Acad. Sect. A 88 (1988), 1–18.
F. Holland and D. Walsh, Distributional inequalities for functions in Besov spaces, Proc. Roy. Irish Acad. Sect. A 94 (1994), 1–17.
M. Jovović and B. MacCluer, Composition operators on Dirichlet spaces, Acta Sci. Math. (Szeged) 63 (1997), 229–247.
H. O. Kim, Derivatives of Blaschke products, Pacific J. Math. 114 (1984), 175–190.
M. Mateljević and M. Pavlović, On the integral means of derivatives of the atomic function, Proc. Amer. Math. Soc. 86 (1982), 455–458.
T. Murai, The value-distribution of lacunary series and a conjecture of Paley, Ann. Inst. Fourier (Grenoble) 31 (1981), 135–156.
D. J. Newman, Interpolation in H ∞, Trans. Amer. Math. Soc. 92 (1959), 501–507.
Ch. Pommerenke, Über die Mittelwerte und Koeffizienten multivalenter Funktionen, Math. Ann. 145 (1961/62), 285–296.
Ch. Pommerenke, Lacunary power series and univalent functions, Michigan Math. J. 11 (1964), 219–223.
Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
L. E. Rubel and R. M. Timoney, An extremal property of the Bloch space, Proc. Amer. Math. Soc. 75 (1979), 45–49.
A. L. Shields, An Analogue of the Fejér-Riesz Theorem for the Dirichlet space, in Conference on Harmonic Analysis in Honor of Antoni Zygmund (Chicago, Ill., 1981), Vol. I, II, Wadsworth, Belmont, CA., 1983, pp. 810–820.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970.
M. Tjani, Compact composition operators on Besov spaces, Trans. Amer. Math. Soc. 355 (2003), 4683–4698.
D. Walsh, A property of univalent functions in A p, Glasgow Math. J. 42 (2000), 121–124.
S. Yamashita, Cowling’s theorem on a Dirichlet finite holomorphic function in the disk, Amer. Math. Monthly 87 (1980), 551–552.
K. Zhu, Analytic Besov spaces, J. Math. Anal. Appl. 157 (1991), 318–336.
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The authors thankfully acknowledge partial support from the following grants. The first author: MTM2008-05561-C02-02 (MICINN) and 2009 SGR 420 (Generalitat de Catalunya). The second author: MTM2007-60854 (MICINN) and FQM-210 and P09-FQM-4468 (Junta de Andalucía). The third author: MTM2009-14694-C02-01 (MICINN). All authors were also partially supported byMTM2008-02829-E (Acciones Complementarias) and Ingenio Mathematica (i-MATH) CSD2006-00032 from MICINN, Spain, and by the Thematic Network “Harmonic and Complex Analysis and Its Applications” (European Networking Programme, ESF).
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Donaire, J.J., Girela, D. & Vukotić, D. On the growth and range of functions in Möbius invariant spaces. JAMA 112, 237–260 (2010). https://doi.org/10.1007/s11854-010-0029-9
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DOI: https://doi.org/10.1007/s11854-010-0029-9