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On \(L^1\)-estimates of derivatives of univalent rational functions. (English) Zbl 1395.30022
Let \(\mathcal{RU}_n\) be the class of univalent rational functions, of degree at most \(n\), in the unit disk \(\mathbb{D}\), without poles in \(\overline{\mathbb{D}}\). Let \(l(R)=\int_\mathbb{T}|R'(z)|dm(z)\); here \(\mathbb{T}=\partial \mathbb{D}\), \(dm(z)=|dz|/(2\pi)\) is the normalized Lebesgue measure on \(\mathbb{T}\), \(R\in\mathcal{RU}_n\). The main aim of the paper is to estimate the value \(\gamma_0:=\limsup_{n\to\infty}\sup_{R\in \mathcal{RU}_n, \|R\|_{\infty,\mathbb{T}\leq 1}} \log l(R)/\log n\) where \(\|R\|_{\infty,\mathbb{T}}=\sup\{|f(z)|: z\in \mathbb{T}\}\). Theorem 1.2 states that \(B_b(1)\leq \gamma_0\leq 1/2\); here \(B_b(t)\) is the integral means spectrum for bounded univalent functions.
The authors also discuss a result by E. P. Dolzhenko [Anal. Math. 4, 247–265 (1978; Zbl 0425.41027)] on estimates of integral means for derivatives of (multivalued) bounded in \(\overline{\mathbb{D}}\) rational functions and give a new shorter proof of the fact. At the final part of the paper, connection with Nevanlinna domains and other related topics are given as well.

MSC:
30C55 General theory of univalent and multivalent functions of one complex variable
30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30J99 Function theory on the disc
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References:
[1] Baranov, A. D.; Fedorovskiy, K. Yu., Boundary regularity of Nevanlinna domains and univalent functions in model subspaces, Sb. Math., 202, 1723-1748, (2011) · Zbl 1254.30039
[2] Baranov, A. D.; Hedenmalm, H., Boundary properties of Green functions in the plane, Duke Math. J., 145, 1-24, (2008) · Zbl 1157.35327
[3] D. Beliaev and S. Smirnov, Harmonic measure on fractal sets, Proceedings of the 4th European Congress of Mathematics European Mathematical Society, Zürich, 2005, pp. 41-59. · Zbl 1079.30026
[4] Carleson, L.; Jones, P. W., On coefficient problems for univalent functions and conformal dimension, Duke Math. J., 66, 169-206, (1992) · Zbl 0765.30005
[5] Carmona, J. J., Mergelyan’s approximation theorem for rational modules, J. Approx. Theory, 44, 113-126, (1985) · Zbl 0574.30041
[6] Carmona, J. J.; Paramonov, P. V.; Fedorovskiy, K. Yu., Uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions, Sb. Math., 193, 1469-1492, (2002) · Zbl 1053.30028
[7] P. J. Davis, The Schwarz Function and its Applications, Math. Assoc. America, Buffalo, NY, 1974. · Zbl 0293.30001
[8] Dolzhenko, E. P., Some exact integral estimates of the derivatives of rational and algebraic functions. applications, Anal. Math., 4, 247-268, (1978) · Zbl 0425.41027
[9] P. L. Duren, Theory of Hp spaces, Academic Press, New York, 1970. · Zbl 0215.20203
[10] E. Dyn’kin, Rational functions in Bergman spaces, Complex Analysis, Operators, and Related Topics, Birkhäuser, Basel, 2000, pp. 77-94. · Zbl 0992.30021
[11] Fedorovskiy, K. Yu., On uniform approximations of functions by n-analytic polynomials on rectifiable contours in C, Math. Notes, 59, 435-439, (1996) · Zbl 0879.30021
[12] Fedorovskiy, K. Yu., Approximation and boundary properties of polyanalytic functions, Proc. Steklov Inst. Math., 235, 251-260, (2001)
[13] Fedorovskiy, K. Yu., On some properties and examples of Nevanlinna domains, Proc. Steklov Inst. Math., 253, 186-194, (2006) · Zbl 1351.30021
[14] J. B. Garnett and D. Marshall, Harmonic Measure, Cambridge University Press, Cambridge, 2005. · Zbl 1077.31001
[15] B. Gustafsson and H. S. Shapiro, What is a quadrature domain? Quadrature Domains and Their Application, Birkhäuser, Basel, 2005, pp. 1-25. · Zbl 1086.30002
[16] Hedenmalm, H.; Shimorin, S., Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Math. J., 127, 341-393, (2005) · Zbl 1075.30005
[17] Hedenmalm, H.; Shimorin, S., On the unversal integral means spectrum of conformal mappings near the origin, Proc. Amer. Math. Soc., 135, 2249-2255, (2007) · Zbl 1109.30003
[18] Kayumov, I. R., On an inequality for the universal spectrum of integral means, Math. Notes, 84, 137-141, (2008) · Zbl 1219.30006
[19] Mazalov, M. Ya., Example of a nonrectifiable Nevanlinna contour, St. Petersburg Math. J., 27, 625-630, (2016) · Zbl 1345.30006
[20] Mazalov, M. Ya.; Paramonov, P. V.; Fedorovskiy, K. Yu., Conditions for the cm-approximability of functions by solutions of elliptic equations, Russian Math. Surveys, 67, 1023-1068, (2012) · Zbl 1262.30027
[21] N. K. Nikolski, Treatise on the Shift Operator, Springer-Verlag, Berlin-Heidelberg, 1986.
[22] Nikolski, N. K., Sublinear dimension growth in the kreiss matrix theorem, Algebra i Analiz, 25, 3-51, (2013)
[23] O’Farrell, A. G., Annihilators of rational modules, J. Funct. Anal., 19, 373-389, (1975) · Zbl 0314.46051
[24] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992. · Zbl 0762.30001
[25] Sola, A., An estimate of the universal means spectrum of conformal mappings, Comput. Methods Funct. Theory, 6, 423-436, (2006) · Zbl 1114.30007
[26] Spijker, M. N., On a conjecture by leveque and trefethen related to the kreiss matrix theorem, BIT, 31, 551-555, (1991) · Zbl 0736.15015
[27] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970. · Zbl 0207.13501
[28] S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI, 2005. · Zbl 1119.37001
[29] Trent, T.; Wang, J. L.-M., Uniform approximation by rational modules on nowhere dense sets, Proc. Amer. Math. Soc., 81, 62-64, (1981) · Zbl 0472.30032
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