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Bohr inequalities via proper combinations for a certain class of close-to-convex harmonic mappings

Molla Basir Ahamed, Partha Pratim Roy

Abstract

Let $ \mathcal{H}(Ω) $ be the class of complex-valued functions harmonic in $ Ω\subset\mathbb{C} $ and each $f=h+\overline{g}\in \mathcal{H}(Ω)$, where $ h $ and $ g $ are analytic. In the study of Bohr phenomenon for certain class of harmonic mappings, it is to find a constant $ r_f\in (0, 1) $ such that the inequality \begin{align*} M_f(r):=r+\sum_{n=2}^{\infty}\left(|a_n|+|b_n|\right)r^n\leq d\left(f(0), \partialΩ\right) \;\mbox{for}\;|z|=r\leq r_f, \end{align*} where $ d\left(f(0), \partialΩ\right) $ is the Euclidean distance between $ f(0) $ and the boundary of $ Ω:=f(\mathbb{D}) $. The largest such radius $ r_f $ is called the Bohr radius and the inequality $ M_f(r)\leq d\left(f(0), \partialΩ\right) $ is called the Bohr inequality for the class $ \mathcal{H}(Ω) $. In this paper, we study Bohr phenomenon for the class of close-to-convex harmonic mappings establishing several inequalities. All the results are proved to be sharp.

Bohr inequalities via proper combinations for a certain class of close-to-convex harmonic mappings

Abstract

Let be the class of complex-valued functions harmonic in and each , where and are analytic. In the study of Bohr phenomenon for certain class of harmonic mappings, it is to find a constant such that the inequality \begin{align*} M_f(r):=r+\sum_{n=2}^{\infty}\left(|a_n|+|b_n|\right)r^n\leq d\left(f(0), \partialΩ\right) \;\mbox{for}\;|z|=r\leq r_f, \end{align*} where is the Euclidean distance between and the boundary of . The largest such radius is called the Bohr radius and the inequality is called the Bohr inequality for the class . In this paper, we study Bohr phenomenon for the class of close-to-convex harmonic mappings establishing several inequalities. All the results are proved to be sharp.
Paper Structure (8 sections, 23 theorems, 79 equations, 5 figures, 6 tables)

This paper contains 8 sections, 23 theorems, 79 equations, 5 figures, 6 tables.

Table of Contents

  1. introduction
  2. Rogosinski radius.
  3. Bohr inequality for harmonic mappings
  4. Bohr inequalities for the class $\mathcal{B}$
  5. Refined versions of the Bohr inequality for the class $\mathcal{B}$
  6. Improved Bohr inequalities for the class $\mathcal{B}$
  7. Main results and corollaries
  8. Proof of Theorems \ref{['th-3.1']} and \ref{['th-3.2']}

Key Result

Theorem 1.1

Abu-CVEE-2010 If $g(z)=\sum_{n=0}^{\infty}b_nz^n\in S(f)$ and $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is univalent, then where $\rho_0^*$ is sharp for Koebe function $f(z)=z/(1-z)^2$.

Figures (5)

  • Figure 1: The figure exhibits the roots $R^{0,0,\cdots0}_{1,\mu, 1,1,1}(M)$ of equation \ref{['eq-3.8']} and the roots $R^{0,0,\cdots0}_{1,\mu, 1,2,1}(M)$ of \ref{['eq-3.9']} respectively for different values of $M$ as defined in Table \ref{['t-1']}
  • Figure 2: The figure exhibits the roots $R^{\frac{16}{9},0,\cdots0}_{0,\mu, 0,m,1}(M)$ of equation \ref{['EQN-3.5']} and the roots $R^{\frac{9}{8},0,\cdots0}_{0,\mu, 0,m,1}(M)$ of \ref{['EQN-3.6']} respectively for different values of $M$ as defined in Table \ref{['T-1']}
  • Figure 3: The figure exhibits the roots of equation \ref{['eq-3.11']} and roots of the equation \ref{['Eq-3.12']} as defined in Table \ref{['t-4']} and Table \ref{['t-5']} respectively.
  • Figure 4: The figure exhibits the roots of equation \ref{['EQN-3.25']} and the roots of equation \ref{['EQN-3.26']} as defined in Table \ref{['T-6']} .
  • Figure 5: The figure exhibits the roots of equation \ref{['eq-3.13']} as defined in Table \ref{['T-7']} .

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.1
  • Definition 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.1
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • ...and 21 more