Extreme Points and Support Points of Families of Harmonic Bloch Mappings
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Extreme Points and Support Points of Families of Harmonic Bloch Mappings Hua Deng1 · Saminathan Ponnusamy2
· Jinjing Qiao1
Received: 12 October 2019 / Accepted: 28 July 2020 / © Springer Nature B.V. 2020
Abstract In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk D. Then we show that a harmonic Bloch mapping f is a support point of the unit ball of the normalized harmonic Bloch spaces in D if and only if the Bloch unit-valued set of f is not empty. We also give a characterization for the support points of the unit ball of the harmonic Bloch spaces in D. Keywords Bloch function · Harmonic Bloch mapping · Extreme point · Support point Mathematics Subject Classification (2010) Primary: 30H30 · 30D45 · 31A05; Secondary: 46E15
1 Introduction and Preliminaries Support points and extreme points of analytic functions play important roles in solving extremal problems. It is known that in the topology of uniform convergence on compacta, any compact family of analytic functions contains support points and the set of all support points contains an extreme point. This remarkable fact plays an active role in solving
Jinjing Qiao
[email protected] Hua Deng [email protected] Saminathan Ponnusamy [email protected] 1
Department of Mathematics, Hebei University, Baoding, Hebei 071002, People’s Republic of China
2
Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India
H. Deng et al.
extremal problems for various families of analytic functions (see [1, 2, 5, 6, 10, 12, 13, 16– 18, 20–22] and for very recent work on this topic, we refer to [14, 15]). The main focus in this article is to extend a number of results from the theory of analytic functions to the case of planar harmonic mappings. In particular, we extend the work of Cima and Wogen [7, Theorem 2] in the setting of little harmonic Bloch mappings, and construct a counterexample to show that [7, Theorem 1 and Corollary 1] fail to hold for (little) harmonic Bloch mappings. Moreover, we establish a characterization for a harmonic Bloch mapping to be a support point of BH,1 which in turn extends the work of Bonk [4, Theorem 3]. The definitions of these mappings and the exact formulation of the results of Cima and Wogen will be addressed later in this section and the results of Bonk in the next section. Let C be the complex plane, and Dr = {z ∈ C : |z| < r} for r > 0. A harmonic mapping f in D = D1 is a complex-valued function of the form f = u + iv, where u and v are real-valued harmonic functions in D. This function has the canonical decomposition f = h + g, where h and g are analytic functions in D, known as analytic and co-anal
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