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On Third-Order Determinant Bounds for the class $\mathcal{S}^*_{B}$

S. Sivaprasad Kumar, Arya Tripathi

Abstract

This paper deals with sharp bounds for the third-order Hankel, Toeplitz and Hermitian-Toeplitz determinant of functions belonging to the class $\mathcal{S}^*_{B}$ of starlike functions associated with a balloon-shaped domain, given by \[ \mathcal{S}^{\ast}_{B}= \left\{ f \in \mathcal{A} : \frac{z f'(z)}{f(z)} \prec \frac{1}{1-\log (1+z)} :=B(z), \quad z \in \mathbb{D} \right\}. \] By applying coefficient inequalities and properties of these functions, we obtain sharp bounds for these determinants. The sharpness of the results is verified by constructing suitable extremal functions.

On Third-Order Determinant Bounds for the class $\mathcal{S}^*_{B}$

Abstract

This paper deals with sharp bounds for the third-order Hankel, Toeplitz and Hermitian-Toeplitz determinant of functions belonging to the class of starlike functions associated with a balloon-shaped domain, given by By applying coefficient inequalities and properties of these functions, we obtain sharp bounds for these determinants. The sharpness of the results is verified by constructing suitable extremal functions.
Paper Structure (3 sections, 4 theorems, 44 equations, 1 figure, 2 tables)

This paper contains 3 sections, 4 theorems, 44 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Main Results

Key Result

Lemma 1

kwon_adam_p1p2lemmalibera_p1p2lemmaneha_third_hankel_willey Let $p \in \mathcal{P}$ and of the form pp. Then for some $\rho, \gamma$ and $\eta$ such that $|\rho| \leq 1, \, |\gamma| \leq 1$ and $|\eta| \leq 1$.

Figures (1)

  • Figure 3: $3D$ plot of $h(p,x)$, for $p\in[0,2],\; x \in [0,1]$.

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof