Bulletin of the
Korean Mathematical Society BKMS

For the classes of analytic functions $f$ defined on the unit disk satisfying $$\frac{2 z {f}'(z)}{f(z) - f(-z)} \prec \varphi(z) \quad \text{and} \quad \frac{(2 z {f}'(z))'}{(f(z) - f(-z))'} \prec \varphi(z) , $$ 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.

Keywords: Univalent functions, starlike functions with respect to symmetric points, fifth coefficient, Hermitian-Toeplitz determinants

MSC numbers: Primary 30C45, 30C50, 30C80

Supported by: The work of the Surya Giri is supported by University Grant Commission, New-Delhi, India under UGC-Ref. No. 1112/(CSIR-UGC NET JUNE 2019).