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Coefficient problems of Starlike Functions Related to a Balloon-Shaped Domain

S. Sivaprasad Kumar, A. Tripathi

Abstract

Recent advances in image and signal processing have drawn on geometric function theory, particularly coefficient estimate problems. Motivated by their significance, we introduce a class of starlike functions related to a balloon-shaped domain \[ \mathcal{S}^*_{\mathcal{B}}= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1}{1-\log(1+z)} := B(z); \; z \in \mathbb{D} \right\}, \] where $B(z)$ maps the unit disk $\mathbb{D}$ onto a balloon-shaped domain. This work establishes bounds for the second order Hankel determinants and second order Toeplitz determinants involving the initial coefficients, the logarithmic coefficients and the logarithmic coefficients of the inverse function for $f \in \mathcal{S}^*_{\mathcal{B}}$

Coefficient problems of Starlike Functions Related to a Balloon-Shaped Domain

Abstract

Recent advances in image and signal processing have drawn on geometric function theory, particularly coefficient estimate problems. Motivated by their significance, we introduce a class of starlike functions related to a balloon-shaped domain where maps the unit disk onto a balloon-shaped domain. This work establishes bounds for the second order Hankel determinants and second order Toeplitz determinants involving the initial coefficients, the logarithmic coefficients and the logarithmic coefficients of the inverse function for
Paper Structure (5 sections, 15 theorems, 94 equations, 1 figure, 1 table)

This paper contains 5 sections, 15 theorems, 94 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Hankel determinant
  4. Toeplitz Determinant
  5. Conclusion

Key Result

Lemma 2.1

cho_kowalczyk_lecko_smiarowska2020: If $p \in \mathcal{P}$ is of the form pp, then for some $\zeta_{1},\;\zeta_{2},\;\zeta_{3}$$\in \mathbb{\overline{D}}$. For $\zeta_{1}\in \mathbb{T}:=\{z \in \mathbb{C}\;;\;|z|=1\}$, there is a unique function $p \in \mathcal{P}$ with $p_1$ as in l1, namely, For $\zeta_1 \in \mathbb{D}$ and $\zeta_2 \in \mathbb{T}$, there is a uniquee function $p\in \mathcal{P

Figures (1)

  • Figure 1: $B(\mathbb{D})$, a balloon-shaped domain, $B(z)=\dfrac{1}{1-\log(1+z)}$.

Theorems & Definitions (26)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 16 more