Sharp bounds of logarithmic coefficient problems for functions with respect to symmetric points

Abstract

The logarithmic coefficients play an important role for different estimates in the theory of univalent functions.
Due to the significance of the recent studies about the logarithmic coefficients, the problem of obtaining the sharp bounds for the second Hankel determinant of these coefficients, that is $H_{2,1}(F_f/2)$ was paid attention. We recall that if $f$ and $F$ are two analytic functions in $\mathbb{D}$, the function $f$ is subordinate to $F$, written $f(z)\prec F(z)$, if there exists an analytic function $\omega$ in $\mathbb{D}$ with $\omega(0)=0$ and $|\omega(z)|<1$, such that $f(z)=F\left(\omega(z)\right)$ for all $z\in\mathbb{D}$. It is well-known that if $F$ is univalent in $\mathbb{D}$, then $f(z)\prec F(z)$ if and only if $f(0)=F(0)$ and $f(\mathbb{D})\subset F(\mathbb{D})$.
A function $f\in\mathcal{A}$ is starlike with respect to symmetric points in $\mathbb{D}$ if
for every $r$ close to $1,$ $r < 1$ and every $z_0$ on $|z| = r$ the angular velocity of $f(z)$
about $f(-z_0)$ is positive at $z = z_0$ as $z$ traverses the circle $|z| = r$ in the positive
direction. In the current study, we obtain the sharp bounds of the second Hankel determinant of the logarithmic coefficients for families $\mathcal{S}_s^*(\psi)$ and $\mathcal{C}_s(\psi)$ where were defined by the concept subordination and $\psi$ is considered univalent in $\mathbb{D}$ with positive real part in $\mathbb{D}$ and satisfies the condition $\psi(0)=1$. Note that $f\in \mathcal{S}_s^*(\psi)$ if
\[
\dfrac{2zf^\prime(z)}{f(z)-f(-z)}\prec\psi(z),\quad z\in\mathbb{D}
\]
and $f\in \mathcal{C}_s(\psi)$ if
\[
\dfrac{2(zf^\prime(z))^\prime}{f^\prime(z)+f^\prime(-z)}\prec\psi(z),\quad z\in\mathbb{D}.
\]
It is worthwhile mentioning that the given bounds in this paper extend and develop some related recent results in the literature. In addition, the results given in these theorems can be used for determining the upper bound of $\left\vert H_{2,1}(F_f/2)\right\vert$ for other popular families.