The hyperbolic metric and an application to growth estimates for holomorphic functions. Iason Efraimidis Department of Mathematics School of Sciences Aristotle University of Thessaloniki

Summary of the Thesis In this thesis, we present a covering theorem of A.Yu. Solynin [21] and the necessary theory for understanding its proof. We begin with P, the class of Carath´eodory, which consists of all holomorphic functions on the unit disk with f (0) = 1 and positive real part. Applying the Schwarz Lemma in P is a simple exercise and yields: |f 0 (0)| ≤ 2

|f (z) − 1| ≤

and

2|z| . 1 − |z|

In 2008, Solynin strenghtened this result, by proving that if we replace the positive real part condition with a weaker geometric property for the image of the function involved, then the above inequalities remain valid. A function f has this geometric property if for all y ∈ R, the line segment {x + yi : 0 ≤ x ≤ 2} intersects with the boundary of the function’s image ∂f (B2 ). Clearly, all functions in P have this property. For the proof of the theorem, necessary tools are the hyperbolic metric, polarization of a region, subordination of two holomorphic functions and Montel’s big Theorem. The thesis starts with the concept of an analytic element and analytic continuation along a curve. We define analytic functions in the generalized sense of Saks and Zygmund [19] and call them general analytic functions. We prove the Monodromy Theorem which states that a general analytic function in a simply connected domain that admits analytic continuation along every curve, reduces to an analytic function. Next, we speak of hyperbolic domains, i.e. domains in C with at least three boundary points, and we prove a useful lemma: f is a holomorphic covering of an hyperbolic domain if-f f −1 admits analytic continuation along every curve. We continue by giving a detailed construction of Dedekind’s elliptic modular function and we use this modular function to prove an extension of Riemann’s Theorem. That is the part of 1

the Uniformization Theorem which states the existence of a general analytic function between any two hyperbolic domains. Following the presentation of Beardon and Minda [4], we gradually define the hyperbolic metric, first on the unit disc, then on simply connected domains and finally on hyperbolic domains. We show the topological equivalence with the Euclidean metric, the completeness of the hyperbolic plane and we study the geodesics at some simple cases. We prove the Principle of the hyperbolic metric, according to which every holomorphic function is either a hyperbolic contraction or a holomorphic covering and a hyperbolic isometry. We continue with Ahlfors’ Lemma [1], which states that the hyperbolic metric is maximal among all C 2 conformal pseudo-metrics with curvature bounded above by -1. We also prove Heins’ strengthened version of Ahlfors’ Lemma, which omits the C 2 requirement and gives a rigid result. We demonstrate the power of this tool by giving a simple proof of Picard, Huber and Liouville theorem. Next, we present the concept of subordination of two holomorphic functions which we denote with f ≺ g. This means that f (0) = g(0) and f = g◦ω for a function ω that satisfies the requirements of Schwarz Lemma. We show that a holomorphic covering is maximal in this sense and prove Littlewood’s and Lindel¨of’s inequalities involving coefficients and p-norms of f and g. We move on to polarization with respect to the real axis which is a geometric transformation that preserves the symmetric part of a set and moves the nonsymmetric part to the upper half plane. In 1997, Solynin [20] proved that the hyperbolic metric descends after this transformation. Next, we prove Montel’s big Theorem using for one more time the elliptic modular function. Finally, we prove Solynin’s Theorem.

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Contents 1. Introduction 2. Analytic continuation 3. The Monodromy Theorem 4. The elliptic modular function 5. The Uniformization Theorem 6. The hyperbolic metric on the unit disk 7. The Schwarz - Pick Lemma 8. The hyperbolic metric on simply connected regions 9. On hyperbolic regions 10. Ahlfors’ Lemma 11. Subordination 12. Polarization 13. Montel’s big Theorem 14. Solynin’s Theorem

References [1] L.V. Ahlfors, An extension of Schwarz’s lemma, Trans. Amer. Math. Soc. 43 (1937), 359-364. [2] L.V. Ahlfors, Complex Analysis, third edition, McGraw-Hill, 1979. [3] L.V. Ahlfors, Conformal Invariants, McGraw-Hill, 1973. [4] A.F. Beardon, D. Minda, The hyperbolic metric and geometric function theory. Proceedings of the international workshop on quasiconformal mappings and their applications, Narosa Publishing House, New Delhi, 2006. [5] D. Betsakos, Polarization, conformal invariants, and Brownian motion, Ann. Acad. Sci. Fenn. Math. 23 (1998), 59-82. [6] H. Chen, On the Bloch constant, in: Arakelian, N. (ed.) et al., Approximation, complex analysis, and potential theory, Kluwer Academic Publishers (2001), 129161. [7] J.B. Conway, Functions of One Complex Variable I, second edition, Springer, 1978. [8] J.B. Conway, Functions of One Complex Variable II, Springer, 1995.

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[9] P. Duren, Univalent Functions, Springer, 1983. [10] W.K. Hayman, P.B. Kennedy, Subharmonic functions vol.I, Academic Press, 1976. [11] W.K. Hayman, Subharmonic functions vol.II, Academic Press, 1989. [12] M. Heins, On a class of conformal metrics, Nagoya Math. J. 21 (1962), 1-60. [13] D. Kraus, O. Roth, Conformal Metrics, to appear at Lecture Notes Series of Ramanujan Math. Society Journal. [14] A.I. Markushevich, Theory of Functions of a Complex Variable, Chelsea Publishing Company, 1977. [15] D. Minda, The strong form of Ahlfors’ Lemma, Rocky Math. J. 17 (1987), 457-461. [16] R. Remmert, Theory of Complex Functions, Springer, 1991. [17] H.L. Royden, The Ahlfors-Schwarz lemma: the case of equality, J. Analyse Math. 46 (1986), 261-270. [18] W. Rudin, Real and Complex Analysis, third edition, McGraw-Hill, 1987. [19] S. Saks and A. Zygmund, Analytic Functions, Monografie Matematyczne, 1952. [20] A.Yu. Solynin, Functional inequalities via polarization, St. Petersburg Math. J. (1997), 1015-1038. [21] A.Yu. Solynin, A Schwarz lemma for meromorohic functions and estimates for the hyperbolic metric, Proc. Amer. Math. Soc. 136 (2008), 3133-3143.

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