Refined Bohr-type inequalities with area measure for bounded analytic functions
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Refined Bohr-type inequalities with area measure for bounded analytic functions Yong Huang1 · Ming-Sheng Liu1
· Saminathan Ponnusamy2
Received: 18 June 2020 / Accepted: 8 September 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we establish five new sharp versions of Bohr-type inequalities for bounded analytic functions in the unit disk by allowing Schwarz function in place of the initial coefficients in the power series representations of the functions involved and thereby, we generalize several related results of earlier authors. Keywords Bohr radius · Bounded analytic functions · Harmonic function · Bohr inequality Mathematics Subject Classification Primary 30A10 · 30C45 · 30C62; Secondary 30C75
1 Introduction and preliminaries Let D := {z ∈ C : |z| < 1} denote the open unit disk in C. A remarkable discovery of Herald Bohr [10] in 1914 states that if H∞ denotes the class of all bounded analytic functions f on D with the supremum norm f ∞ := supz∈D | f (z)|, then B0 ( f , r ) := |a0 | +
∞
|an |r n ≤ f ∞ for 0 ≤ r ≤ 1/6,
n=1
B
Ming-Sheng Liu [email protected] Yong Huang [email protected] Saminathan Ponnusamy [email protected]
1
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, Guangdong, China
2
Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India 0123456789().: V,-vol
(1.1)
50
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where ak = f (k) (0)/k! for k ≥ 0. Later M. Riesz, I. Shur and F. W. Wiener, independently proved its validity on a wider interval 0 ≤ r ≤ 1/3, and the family of functions ϕa (z) = (a − z)/(1 − az) (|a| < 1) as a → 1 demonstrates that the number 1/3 is optimal. This result is usually referred to as Bohr’s power series theorem for the unit disk and 1/3 is called the Bohr radius. We refer the paper of Bohr [10] which contains the proof of Wiener showing that the Bohr radius is 1/3. See also [30,31] for other proofs. Then it is worth pointing out that there is no extremal function in B such that the Bohr radius is precisely 1/3 (cf. [7], [13, Corollary 8.26] and [15]). Several aspects of Bohr’s inequality and its extensions in various settings may be seen in the literature. For example, the Bohr radius for analytic functions from the unit disk into different domains, such as the punctured unit disk or the exterior of the closed unit disk or concave wedge-domains, have been analyzied in [1–4]. Ali et al. [6,15] considered the problem of determining Bohr radius for the classes of even and odd analytic functions and for alternating series. The articles [7,19,24] concerned with the class of all sense-preserving harmonic mappings and the Bohr radius for sensepreserving harmonic quasiconformal mappings. Defant [11] improved a version of the Bohnenblust–Hille inequality, and in 2004, Paulsen [26] proved a uniform algebra analogue of the classical inequality of Bohr concerning Fourier coefficients of bounded holomorphic functions. In [25,27], the authors demonstrated the classical Bohr inequality using different me
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