<p>The primary objective of this paper is to establish several sharp results concerning the Bohr inequality, the refined Bohr inequality, and the improved Bohr inequality for the classes of analytic functions and harmonic mappings defined on the shifted disks <Equation ID="Equ14"> <EquationSource Format="TEX">$\begin{aligned} \Omega _{\gamma }=\left\{ z\in \mathbb {C}:\left| z+\frac{\gamma }{1-\gamma }\right| <\frac{1}{1-\gamma }\right\} \quad \text {for}\quad \gamma \in [0,1).\end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi mathvariant="normal">Ω</mi> <mi>γ</mi> </msub> <mo>=</mo> <mfenced close="}" open="{"> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mfenced close="|" open="|"> <mi>z</mi> <mo>+</mo> <mfrac> <mi>γ</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>γ</mi> </mrow> </mfrac> </mfenced> <mo><</mo> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>-</mo> <mi>γ</mi> </mrow> </mfrac> </mfenced> <mspace width="1em" /> <mtext>for</mtext> <mspace width="1em" /> <mi>γ</mi> <mo>∈</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation></p>

Bohr Phenomenon for Analytic and Harmonic Mappings on Shifted Disks

期刊论文
发表日期：

2026年7月14日

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Vasudevarao Allu,
Raju Biswas,
Rajib Mandal

摘要

The primary objective of this paper is to establish several sharp results concerning the Bohr inequality, the refined Bohr inequality, and the improved Bohr inequality for the classes of analytic functions and harmonic mappings defined on the shifted disks $\begin{aligned} \Omega _{\gamma }=\left\{ z\in \mathbb {C}:\left| z+\frac{\gamma }{1-\gamma }\right| <\frac{1}{1-\gamma }\right\} \quad \text {for}\quad \gamma \in [0,1).\end{aligned}$ $\begin{matrix} Ω_{γ} = \{z \in C : |z + \frac{γ}{1 - γ}| < \frac{1}{1 - γ}\} for γ \in [0, 1) . \end{matrix}$