Properties of normal harmonic mappings
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Properties of normal harmonic mappings Hua Deng1 · Saminathan Ponnusamy2
· Jinjing Qiao1
Received: 12 October 2019 / Accepted: 14 August 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract In this paper, we present several necessary and sufficient conditions for a harmonic mapping to be normal. Also, we discuss maximum principle and five-point theorem for normal harmonic mappings. Furthermore, we investigate the convergence of sequences for sense-preserving normal harmonic mappings and show that the asymptotic values and angular limits are identical for normal harmonic mappings. Keywords Normal functions · Normal harmonic mappings · Spherical derivative · Maximum principle Mathematics Subject Classification Primary 30D45 · 31A05; Secondary 30G30 · 30H05
1 Introduction and main results Let D = {z ∈ C : |z| < 1} denote the unit disk in the complex plane C. A function f meromorphic in D is called a nor mal f unction if the family F = { f ◦ϕ : ϕ ∈ Aut(D)} is a normal family, where Aut(D) denotes the class of conformal automorphisms of D (cf. [10]). Normal functions were first studied by Yosida [17]. Subsequently, Noshiro
Communicated by Adrian Constantin. The research of this paper is supported by NSF of Hebei Science Foundation (No. A2018201033).
B
Jinjing Qiao [email protected] Hua Deng [email protected] Saminathan Ponnusamy [email protected]
1
Department of Mathematics, Hebei University, Baoding 071002, Hebei, People’s Republic of China
2
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
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H. Deng et al.
[13] gave a characterization of normal functions by showing that a meromorphic function f is normal if and only if sup(1 − |z|2 ) f # (z) < ∞,
(1.1)
z∈D
where f # denotes the spherical derivative of f given by f # (z) = | f (z)|/(1 + | f (z)|2 ). The condition (1.1) is equivalent to say that f is Lipschitz when regarded as a function from the hyperbolic disk D into the extended complex plane endowed with the chordal distance (cf. [10]) which is defined as follows: The chor dal distance χ (a, b) between the complex values a and b, considered as points on the Riemann sphere, is given by ⎧ 0 if a = b, ⎪ ⎪ ⎪ |a − b| ⎪ ⎨ if a = ∞ = b, χ (a, b) = (1.2) 1 + |a|2 1 + |b|2 ⎪ ⎪ 1 ⎪ ⎪ if a = ∞ = b. ⎩ 1 + |a|2 Normal functions play important roles in studying properties of meormorphic functions, specially the behaviour in the boundary of meormorphic functions. Many results have appeared in the literature, see, for example, [8–10,12,14,16]. The main focus in this article is to extend a number of results from theory of analytic functions to the case of planar harmonic mappings. Let be a simply connected domain in C. A harmonic mapping f on is a complex-valued function which has the canonical decomposition f = h + g, where h and g are analytic in and g(z 0 ) = 0 at some prescribed point z 0 ∈ . We recall that (see [11]) a necessary and sufficient condition for a complex-valued harmonic mapping f = h + g is locally univalent and sense-pres
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