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

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