Bohr's inequalities for the analytic functions with lacunary series and harmonic functions

By Ilgiz R. Kayumov, Saminathan Ponnusamy. Journal of Mathematical Analysis and Applications, 2018

126 citations recorded by the source corpus.

Abstract We determine the Bohr radius for the class of all functions f of the form f ( z ) = z m ∑ k = 0 ∞ a k p z k p analytic in the unit disk | z | 1 and satisfy the condition | f ( z ) | ≤ 1 for all | z | 1 . In particular, our result also contains a solution to a recent conjecture of Ali et al. [9] for the Bohr radius for odd analytic functions, solved by the authors in [17] . We consider a more flexible approach by introducing the p -Bohr radius for harmonic functions which in turn contains the classical Bohr radius as special case. Also, we prove several other new results and discuss p -Bohr radius for the class of odd harmonic bounded functions.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/323a2099540b6389ae84f409246ca9d751793dcf

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Citations in the corpus

Bohr phenomenon for certain subclasses of harmonic mappings (2020, background): “locally univalent harmonic mappings, k-quasiconformal mappings, bounded harmonic functions, lacunary series have been studied extensively in [23, 24].”
Bohr radius for certain classes of starlike and convex univalent functions (2020, background): “Bohr phenomenon for harmonic mappings has also been extensively studied by several authors (see [4, 11, 17, 23, 24, 26, 30]).”
Sharp Bohr type inequality (2020, methodology): “See [25, 26, 28] for new proofs of it in a general form.”
On a powered Bohr inequality (2018, methodology): “71] What is the exact value of the (powered) Bohr radius ρp, p ∈ (1, 2)? Is it true that ρp = rp? Using the method of proofs of our recent approach from [12,13], we solve this problem affirmatively in the following form.”
Improved Bohr's phenomenon in quasi-subordination classes (2019, result): “See also recent works from [3,4,6,9,11,13–17] and the references therein.”
Refined Bohr inequality for bounded analytic functions (2020, background): “In order to determine the Bohr radius for the class of odd functions in the family B, which was posed in [6], the authors in [19, 20] established a more general result consisting of functions of the form fp,m(z) = ∞ n=0 apn+mz pn+m in B.”
On the Bohr inequality with a fixed zero coefficient (2019, methodology): “Several improved versions of the classical Bohr’s inequality were given by Kayumov and Ponnusamy in [19] (see also [20]) whereas Evdoridis et al.”
Bohr Radius for Subordination and K-quasiconformal Harmonic Mappings (2019, result): “Some of the recent results from [4, 13–15] are included in the latest two surveys.”