Bohr type inequality for certain integral operators and Fourier transform on shifted disks

Abstract

In this paper, we derive the sharp Bohr type inequality for the Cesáro operator, Bernardi integral operator, and discrete Fourier transform acting on the class of bounded analytic functions defined on shifted disks \beas Ω_γ=\left\{z\in\mathbb{C}:\left|z+\fracγ{1-γ}\right|<\frac{1}{1-γ}\right\}\quad\text{for}\quadγ\in[0,1).\eeas

Figures (3)

• Figure 1: The graphs of $C_{\gamma}$ when $\gamma=0,0.2,0.4,0.5,0.7$
• Figure 2: The graph of $-2 \left(1/(1-\rho)+\log(1-\rho)/\rho\right)$ for $\rho\in[0,1)$
• Figure 3: The graph of $3(1-\rho)\log(1-\rho)+2\rho$ for $\rho\in[0,1)$

Theorems & Definitions (11)

• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.1
• Remark 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.2
• proof : Proof of Theorem \ref{['Th1']}
• proof : Proof of Theorem \ref{['Th2']}