AUTHORS: Ronen Peretz
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ABSTRACT: In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization functions. We mostly deal with the Steiner symmetrization G of an analytic function f in the unit disk U. We pose few problems we can not solve. An intriguing one is that of the inequality Z2π 0 |f(reiθ)| p dθ ≤ Z2π 0 |G(reiθ)| p dθ, 0 < p < ∞ which is true for p = 2 but can not be true for too large p. What is the largest such exponent or its supremum?
KEYWORDS: circular symmetrization, Steiner symmetrization, extremal problems.
REFERENCES:
[1] Alexander, H., Taylor, B. A. and Ullman, J. L., Areas of projections of analytic sets, Invent. Math., Volume 16, 1972, pp. 335-341.
[2] A. Baernstein II, Integral means, univalent functions and circular symmetrization, Acta Mathematica, Volume 133, Issue 1, 1974, pp. 139-169.
[3] Betsakos, Dimitrios, Lindelof’s principle and ¨ estimates for holomorphic functions involving area, diameter, or integral means, computational Methods and Function Theory,Volume 14, issue 1, 2014, pp. 85-105.
[4] Bieberbach, L., Uber einen Satz des Herrn ¨ Caratheodory, ´ Nachr. Akad. Wiss. Gonttingen. ¨ 4, 1913, pp. 552-561.
[5] L. de Branges, A proof of the Bieberbach conjecture, Acta Mathematica, Volume 154, 1985, pp. 137-152.
[6] Caratheodory, C., Conformal representation, ´ Cambridge at the University press, no. 28, 1969.
[7] Caratheodory, C., Untersuchungen ´ uber ¨ die Konforme Abbildung von festen und veranderlichen Gebieten, ¨ Math. Ann. 72, 1912, pp. 107-144.
[8] Duren, P. L., Theory of Hp spaces, Pure and Applied Mathematics 38, Academic Press, 1970.
[9] Peter Duren and Glenn Schober,Nonvanishing Univalent Functions*, Math. Z., Volume 170, 1980, pp. 195-216.
[10] Matts Essen and Daniel F. Shea, On some ques- ´ tions of uniqueness in the theory of symmetrization, Annales AcademiæScientiarum Fennicæ, Series A. I. Mathematica, Volume 4, 1978/1979, pp. 311-340.
[11] Goluzin, G. M., Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Volume 26, American Mathematical Society, Providence, Rhode Island 02904, 1969.
[12] Hayman, W. K., Multivalent Functions, Cambridge Tracts in Mathematics 110, Cambridge University Press, second edition, 1994.
[13] Hille, E., Analytic Function Theory, Volume II., Boston, 1962.
[14] Kobayashi, S., Image areas and H2 norms of analytic functions, Proc. Amer. Math. Soc., Volume 91, 1984, pp. 315-320.
[15] Littlewood, J. E., On inequalities in the theory of functions, Proc. London Math. Soc., Volume 23, 1925, pp. 481-519.
[16] Markusevi ˇ c, A. I., Sur la repr ˇ esentation con- ´ forme des domaines a fronti ´ eres, ´ Mat. Sb. 1(43), 1936, pp. 863-886.
[17] Rado, T., Sur la repr ´ estations conforme de do- ´ maines variables, Acta Szeged 1, 1922-1923, pp. 180-186.
[18] Makoto Sakai, Isoperimetric inequalities for the least harmonic majorant of |x| p , Transactions of the American Mathematical Society, Volume 299, Number 2, 1987, pp. 431-472.