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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: SCHWARZ-PICK-TYPE ESTIMATES FOR THE HYPERBOLIC DERIVATIVE | SCHWARZ-PICK-TYPE ESTIMATES FOR THE HYPERBOLIC DERIVATIVE PETER R. MERCER Received 12 February 2005 Revised 3 November 2005 Accepted 8 November 2005 We obtain Schwarz-Pick-type estimates for the hyperbolic derivative of an analytic selfmap of the unit disk in C. Copyright 2006 Peter R. Mercer. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Preliminaries We denote by A the open unit disk in C and for z e A we denote by ộz e Aut A the automorphism which interchanges 0 and z ộz A z - A 1 - zA . We denote by p the hyperbolic distance on A p A z tanh-1 ộz A I 1 log 1 z A I . 1.1 2 1 I z A I The following is a well-known consequence of the maximum principle. Schwarz s Lemma 1.1. Let f A A be analytic with f 0 0. Then I f A A that is p f A f 0 p A 0 VA e A. 1.2 Consequently we have also f 0 1. To remove the normalization f 0 0 one may consider the function g f f ệz 1.3 which has f z 1 - z 2 g 0 0 g 0 Ị y 1.4 1 - f z l to obtain the following. Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 96368 Pages 1-6 DOI 10.1155 JIA 2006 96368 2 Schwarz-Pick-type estimates for the hyperbolic derivative Schwarz-Pick Lemma 1.2. Let f A A be analytic. Then ộf z f A I ộz A I that is p f A f z p A z VA z e A. 1.5 Consequently f z g 0 has I f z 1 and so p f z is defined on A as long as f is not an automorphism for in this case I f I 1. As such we are interested in the following two results. Theorem 1.3 see 6 . Let f A A be analytic and not an automorphism. Then p 0 f A - p 0 f z 2p A z VA z e A. 1.6 So for example if f A and f z are on the same side of a ray emanating from the origin then p f A f z 2p A z . Theorem 1.4 see 1 . Let f A A be analytic not an automorphism with f 0 0. Then p f 0 f z 2p 0 z Vz e A. 1.7 In the next section of this paper we employ a procedure which yields
