Schwarz Lemma, and Distortion for Harmonic Functions Via Length and Area
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Schwarz Lemma, and Distortion for Harmonic Functions Via Length and Area M. Mateljevi´c1 Received: 26 July 2018 / Accepted: 22 August 2019 / © Springer Nature B.V. 2020
Abstract We give sharp estimates for distortion of harmonic mappings u from the unit disc U into Rm , at a prescribed point by means of diameter and area of the corresponding surface S = u(U), and via the generalized length of the boundary of S. Keywords Harmonic and holomorphic functions · Hyperbolic distance · Schwarz lemma · Quasiconformal mappings · Diametar · Length · Area Mathematics Subject Classification (2010) Primary 30C80 · Secondary 30F45
1 Introduction and Basic Definitions Motivated by the role of Schwarz Lemma in complex analysis and some recent results, in 2016, cf. [15](a), the author has posted the current Research project Schwarz lemma, the Carath´eodory and Kobayashi Metrics and Applications in Complex Analysis.1 Various discussions regarding the subject can also be found in the Q&A section on Researchgate under the question ” What are the most recent versions of The Schwarz Lemma ?”, [15] (b). During the fall semester 2017 at Belgrade seminar [13], we have communicated about Schwarz lemma and we have posted the arXiv paper [12], in which we have considered various version of Schwarz lemma and its relatives related to harmonic and holomorphic functions including distortion of harmonic mappings, and several variables. For the results of [12] see also [14] and for further results [17, 18]. In [18] we give review of some recent author results which are related to Schwarz type inequalities for functions admitting a general Poisson type representation. This paper is continuation of these research. During my work on the subject, D. Kalaj gave an interesting communication, cf. [6](from which we have learned about his arXiv papers [4, 5]), and immediately we have realized
1 Motivated
by S. G. Krantz paper [9]
M. Mateljevi´c
[email protected] 1
Faculty of mathematics, University of Belgrade, Studentski Trg 16, Belgrade, Serbia
M. Mateljevi´c
that we can adapt our previous consideration to connect with his work. In particular, using different approach we can give new insight to these results as well as further results. We first need some definitions. Definition 1 a1) By C we denote the complex plane, by U the unit disk and by T the unit circle. By ∂G or bG we denote the boundary of G. If f : U → Rm and 0 ≤ r < 1 we define the function fr by fr (z) = f (rz). a2) For a function h and complex coordinate z = x + iy ∈ C, x, y ∈ R, we use notation D1 h = hx and D2 h = hy for partial derivatives; ∂h = 12 (hx − ihy ) and ∂h = c c 1 c c 2 (hx +ihy ); we also use notations Dh = Dz h = D1 h and Dh = D z h = D 1 h instead of ∂h and ∂h respectively when it seems convenient. For α, 0 ≤ α < 2π , f : U → Rm and z ∈ U we denote by Dα f (z) the directional derivative of f at z Dα f (z) := fx (z) cos α + fy (z) sin α. We then define f (z) and λf (z) as follows: f (z) = maxα |Dα f (z)|, λf (z) = minα |Dα f (z)|. It is convenient to i
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