Logarithmic coefficients play a crucial role in the theory of univalent functions. In this study, we focus on the classes \({\mathcal {S}}_e^*\) and \({\mathcal {C}}_e\) of starlike and convex functions, respectively, \( {\mathcal {S}}_e^*{:}{=} \left\{ f \in {\mathcal {S}} : \frac{zf'(z)}{f(z)} \prec e^z, \ z \in \mathbb {D} \right\} ,\) and \( \mathcal {C}_e {:}{=} \left\{ f \in \mathcal {S} : 1 + \frac{z f''(z)}{f'(z)} \prec e^z, \ z \in \mathbb {D} \right\} .\) This paper investigates the sharp bounds of the logarithmic coefficients and the Hermitian-Toeplitz determinant of these coefficients for the classes \(\mathcal {S}_e^*\) and \(\mathcal {C}_e\) . Additionally, we examine the generalized Zalcman conjecture and the generalized Fek ete-Szegö inequality for these classes \(\mathcal {S}_e^*\) and \(\mathcal {C}_e\) and show that the inequalities are sharp.
