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

For the classes of analytic functions $f$ defined on the unit disk satisfying $$\frac{z{f}^{\mathrm{\prime }}(z)}{f(z)-f(-z)}\prec \phi (z)\text{and}\frac{(2z{f}^{\mathrm{\prime }}(z){)}^{\mathrm{\prime }}}{(f(z)-f(-z){)}^{\mathrm{\prime }}}\prec \phi (z),$$ denoted by ${\mathcal{S}}_s^{\ast }(\phi )$ and ${\mathcal{C}}_s(\phi )$ 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.