Bohr-type inequalities of analytic functions

Liu, Ming-Sheng; Shang, Yin-Miao; Xu, Junfeng

doi:10.1186/s13660-018-1937-y

Cited by 33 publications

(17 citation statements)

References 16 publications

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“…Furthermore, Kayumov and Ponnusamy [24] introduced and studied the Bohr-Rogosinski inequality and found the Bohr-Rogosinski radius. Based on the initiation of [23], several forms of Bohr-type inequalities for the family B were considered in [31] when the Taylor coefficients of classical Bohr inequality are partly or completely replaced by higher order derivatives of f . Let S r (f ) denote the area of the image of the subdisk |z| < r under the mapping f and when there is no confusion, we let for brevity S r for S r (f ).…”

Section: Theorem B ([37mentioning

confidence: 99%

The Bohr-type operator on analytic functions and sections

Huang¹,

Liu²,

Ponnusamy³

2021

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In this paper, firstly we prove two refined Bohr-type inequalities associated with area for bounded analytic functions f (z) = ∞ n=0 a n z n in the unit disk. Later, we establish the Bohr-type operator on analytic functions and sections. Preliminaries and some basic questionsThere has been a intensive research activity on Bohr's phenomenon, examined first in 1914 by Bohr [14]. The main purpose of this article is to continue the investigation on the classical Bohr inequality in the refined formulation studied recently in [20,38] for the case of analytic functions bounded in the unit disk. See the recent survey articles [6,19, 26] and [17, Chapter 8]. Bohr's idea naturally extends to functions of several complex variables. The interest in the Bohr phenomena was revived in the nineties due to extensions to holomorphic functions of several complex variables and to more abstract settings. The Bohr radius for analytic functions from the unit disk into special domains (eg. the punctured unit disk, the exterior of the closed unit disk, and concave wedgedomains) have been discussed in [1,2,3,5]. Ali et al. [9] considered Bohr's phenomenon for even and odd analytic functions and for alternating series. This study was continued by Kayumov and Ponnusamy [21,22], which in turn settled one of the conjectures, proposed in [9], on Bohr radius for odd analytic functions. In continuation of the investigation on this topic, the authors in [10,25,33] concerned the Bohr radius for the class of all sense-preserving harmonic mappings and sense-preserving K-quasiconformal harmonic mappings. In [34,36], authors demonstrated the classical Bohr inequality using different methods of operators. Several other aspects and generalizations of Bohr's inequality may be obtained from [15,16,21,32,35] and the references therein for some detailed account of work on this topic. In particular, after the appearance of the articles by Abu Muhanna et al. [6] and, Kayumov and Ponnusamy [22], several investigations and new problems on Bohr's inequality in the unit disk case have appeared in the literature (cf. [4, 11, 18, 23, 29, 31, 37, 38]). 1.1. Classical Inequality of H. Bohr. Let A denote that class of analytic functions f (z) = ∞ n=0 a n z n in the unit disk D := {z ∈ C : |z| < 1} and B = {f ∈ A : |f (z)| ≤ 1 in D}. For a fixed z ∈ D, let F z = {f (z) : f ∈ A} and introduce the Bohr

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Section: Theorem B ([37mentioning

confidence: 99%

The Bohr-type operator on analytic functions and sections

Huang¹,

Liu²,

Ponnusamy³

2021

Preprint

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“…Therefore in Section 3, we mainly extend the Bohr radius for the family P. Moreover, the properties of M f (r) are extended to the alternating series A f (r), which was originally investigated in [4] by Ali et al In fact they have shown that if f ∈ B, then |A f (r)| < 1 holds for r ≤ 1/ √ 3. In addition, according to Liu et al in [26], the Bohr radius can be generalized by using the property of A f (r). Surprisingly, in recent papers, the authors in [17,24,33,34] refined the Bohr inequality in improved form by replacing…”

Section: Introductionmentioning

confidence: 99%

“…Known important theorems. Recently, Kayumov and Ponnusamy et al [21] (see also [32]) refined Theorem A in the following form whereas, Liu et al [26] proved the estimate of the alternating series.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2, we mainly state and prove several key lemmas which play key role in the proof of our main Theorems. In Section 3, we present partial answers to Problem 1 and present other promotion concerning functions f ∈ P. Moreover, in Section 4, we also discuss and refine the Bohr-type inequalities of the alternating series A f (r) and improve some conclusions of [4] and [26]. Throughout the article for a ∈ [0, 1), we let…”

Section: Introductionmentioning

confidence: 99%

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Generalization of Bohr-type inequality in analytic functions

Lin,

Liu,

Ponnusamy

2021

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This paper mainly uses the nonnegative continuous function {ζ n (r)} ∞ n=0 to redefine the Bohr radius for the class of analytic functions satisfying Re f (z) < 1 in the unit disk |z| < 1 and redefine the Bohr radius of the alternating series A f (r) with analytic functions f of the form f (z) = ∞ n=0 a pn+m z pn+m in |z| < 1. In the latter case, one can also get information about Bohr radius for even and odd analytic functions. Moreover, the relationships between the majorant series M f (r) and the odd and even bits of f (z) are also established. We will prove that most of results are sharp.

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“…Recently, the cases p = 1 and N = 2 were considered in [9]. Also, the situations m = 1 and p = 1 or 2 were investigated in [32]. For p = 1, 2 and N = 5, 10, 15, the values R m,N 8 (p) are computed in Table 8.…”

mentioning

confidence: 99%