Sharp Bounds of Fifth Coefficient and Hermitian-Toeplitz determinants for Sakaguchi Classes

TL;DR

This paper establishes sharp bounds for the fifth Taylor coefficient and third order Hermitian Toeplitz determinants for specific classes of analytic functions defined by subordination conditions, extending known results.

Contribution

It provides the first sharp bounds for the fifth coefficient and Hermitian Toeplitz determinants for the classes al{S}^*_s(i) and al{C}_s(i), advancing the understanding of these function classes.

Findings

Sharp bound of the fifth Taylor coefficient obtained.

Sharp estimates for third order Hermitian Toeplitz determinants established.

Results lead to new and known bounds for these classes.

Abstract

For the classes of analytic functions $f$ defined on the unit disk satisfying $$\frac{z {f}'(z)}{f(z) - f(-z)} \prec \varphi(z) \quad \text{and} \quad \frac{(2 z {f}'(z))'}{(f(z) - f(-z))'} \prec \varphi(z),$$ denoted by $\mathcal{S}^*_s(\varphi)$ and $\mathcal{C}_s(\varphi)$ respectively, the sharp bound of the $n^{th}$ Taylor coefficients are known for $n=2,$ $3$ and $4$. In this paper, we obtain the sharp bound of the fifth coefficient. Additionally, the sharp lower and upper estimates of the third order Hermitian Toeplitz determinant for the functions belonging to these classes are determined. The applications of our results lead to the establishment of certain new and previously known results.