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

Self Cite

We consider the class of all sense‐preserving harmonic mappings f=h+g¯ of the unit disk double-struckD, where h and g are analytic with g(0)=0, and determine the Bohr radius if any one of the following conditions holds: 1.h is bounded in double-struckD. 2.h satisfies the condition Re h(z)≤1 in double-struckD with h(0)>0. 3.both h and g are bounded in double-struckD. 4.h is bounded and g′false(0false)=0. We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in double-struckD is strictly less than 1. In addition, we determine the Bohr radius for the space scriptB of analytic Bloch functions and the space BH of harmonic Bloch functions. The paper concludes with two conjectures.

“…As in the symmetric case of analytic functions (see [2,11,12]), we have the following analog result for harmonic functions.…”

Section: Introductionmentioning

confidence: 85%

“…Thus, by combining the resulting inequality with the last inequality, we find that

0true \sum_{n = 1}^{\infty} (| a_{n} | + | b_{n} |) r^{n} \leq \frac{2 r}{\sqrt{1 - r^{2}}} \leq 1 for r \leq \frac{1}{\sqrt{5}} .

Although this simple approach gives a good estimate, the number

1 / \sqrt{5}

Section: The Proofs Of Theorems Andmentioning

confidence: 99%

Section: The Proofs Of Theorems 11 12 13 16 and 17mentioning

confidence: 99%

“… for the importance, background, and several other recent results and extensions. For certain recent results, see .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Bohr radius for locally univalent harmonic mappings

Kayumov

Ponnusamy

Shakirov

2018

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Improved Bohr inequality for harmonic mappings

Liu¹,

Ponnusamy²

2022

Self Cite

In order to improve the classical Bohr inequality, we explain some refined versions for a quasi‐subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk D${\mathbb {D}}$. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form f=h+g¯$f=h+\overline{g}$, where gfalse(0false)=0$g(0)=0$, the analytic part h is bounded by 1 and that |g′false(zfalse)|≤k|h′false(zfalse)|$|g^{\prime }(z)|\le k|h^{\prime }(z)|$ in D${\mathbb {D}}$ and for some k∈false[0,1false]$k\in [0,1]$.

Bohr's inequality for holomorphic and pluriharmonic mappings with values in complex Hilbert spaces

Hamada

2023

Let double-struckBH$\mathbb {B}_H$ be the unit ball of a complex Hilbert space H. First, we give a Bohr's inequality for the holomorphic mappings with lacunary series with values in complex Hilbert balls. Next, we give several results on Bohr's inequality for pluriharmonic mappings with values in ℓ2. Note that the Bohr phenomenons that we have obtained are completely different from those in the case with values in C$\mathbb {C}$ and are sharp in the case with values in ℓ2.