Lect Rasila Iwhmhm09[1]

International Workshop on Harmonic Mappings and Hyperbolic Metrics, IIT Madras, India, 2009

MY LECTURES IN IWHMHM09 Antti Rasila Helsinki University of Technology (to become Aalto University School of Science and Technology in the beginning of 2010), Espoo, Finland; [email protected]

My lectures will be concerned with the following topics: 1. Introduction to quasiconformal mappings in the complex plane. 2. Distortion results for quasiconformal, quasiregular and harmonic mappings. 3. Lindel¨of-type boundary behavior results for certain classes of functions. In the first lecture, an overview of the theory of quasiconformal mappings in the plane will be given. The focus is in the basic definitions and properties such as the conformal modulus. This lecture is intended to give preliminary information to the following ones. Then, in the next lecture we will deal with distortion results for the above classes of functions. The main topic here is the Schwarz lemma and its various versions. The connection between multiplicity of the function at a given point, and the distortion will also be discussed. Finally, we will study some generalizations of the so-called Lindel¨of theorem for other classes of mappings. In its classical form, this result gives a connection between the sequential and non-tangential limits of a bounded analytic function. In particular, the connection between multiplicities of the zeros and boundary behavior of bounded harmonic and quasiregular mappings will be discussed. MAIN REFERENCES 1. D. Bshouty, W. Hengartner, “Univalent harmonic mappings in the plane,” in: Handbook of complex analysis: geometric function theory (Edited by K¨ uhnau), Vol. 2, Elsevier, Amsterdam, 2005, pp. 479–506. 2. P. Duren, Harmonic Mappings in the Plane, Cambridge University Press, Cambridge (2004). 3. O. Lehto, K.I. Virtanen, Quasiconformal mappings in the plane, Springer, Berlin (1973). 4. S. Ponnusamy, A. Rasila, “On zeros and boundary behavior of bounded harmonic functions,” Analysis (Munich), Forthcoming, (2009). 5. A. Rasila, “Multiplicity and boundary behavior of quasiregular maps,” Math. Z., 250, No. 3, 611–640 (2005).

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