Logarithmic coefficients play a crucial role in the theory of univalent functions. In this study,we focus on the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ of starlike and convex functions, respectively,
\begin{align*}
\mathcal{S}_e^\ast := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec e^z, \ z \in \mathbb{D} \right\}, \end{align*} and \begin{align*}
\mathcal{C}_e := \left\{ f \in \mathcal{S} : 1 + \frac{z f''(z)}{f'(z)} \prec e^z, \ z \in \mathbb{D} \right\}. \end{align*}
This paper investigates the sharp bounds of the logarithmic coefficients and the Hermitian-Toeplitz determinant of these coefficients for the classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$. Additionally, we examine the generalized Zalcman conjecture and the generalized Fekete-Szegö inequality for these classes $\mathcal{S}_e^\ast$ and $\mathcal{C}_e$ and show that the inequalities are sharp.
We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions $f$ with $f/g\in\mathcal{P}$ and $g/(zp)\in\mathcal{P}$ for some normalized analytic function $g$ and $p\in \mathcal{P}$ . The second class is defined by replacing the condition $f/g\in\mathcal{P}$ by $|(f/g)-1|<1$ while the other class consists of normalized analytic functions $f$ with $f/(zp)\in\mathcal{P}$ for …
