Abstract
In our present study, two subclasses of starlike functions which are symmetric about the origin are considered. These two classes are defined with the use of the sigmoid function and the trigonometric function, respectively. We estimate the first four initial logarithmic coefficients, the Zalcman functional, the Fekete-Szego functional, and the bounds of second-order Hankel determinants with logarithmic coefficients for the first class S-seg* and improve the obtained estimate of the existing second-order Hankel determinant of logarithmic coefficients for the second class S-sin*. All the bounds that we obtain in this article are proven to be sharp.