Coefficient bounds for starlike functions associated with Gregory coefficients

Abstract

It is of interest to know the sharp bounds of the Hankel determinant, Zalcman functionals, Fekete-Szeg$ \ddot{o} $ inequality as a part of coefficient problems for different classes of functions. Let $\mathcal{H}$ be the class of functions $ f $ which are holomorphic in the open unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ of the form \begin{align*} f(z)=z+\sum_{n=2}^{\infty}a_nz^n\; \mbox{for}\; z\in\mathbb{D} \end{align*} and suppose that \begin{align*} F_{f}(z):=\log\dfrac{f(z)}{z}=2\sum_{n=1}^{\infty}\gamma_{n}(f)z^n, \;\; z\in\mathbb{D},\;\;\log 1:=0, \end{align*} where $ \gamma_{n}(f) $ is the logarithmic coefficients. The second Hankel determinant of logarithmic coefficients $H_{2,1}(F_{f}/2)$ is defined as: $H_{2,1}(F_{f}/2) :=\gamma_{1}\gamma_{3} -\gamma^2_{2}$, where $\gamma_1, \gamma_2,$ and $\gamma_3$ are the first, second and third logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we first establish sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/64$ with logarithmic coefficients for the classes of starlike functions associated with Gregory coefficients. In addition, we establish the sharpness of Fekete-Szeg$ \ddot{o} $ inequality, Zalcman functional and generalized Zalcman functional for the class starlike functions associated with Gregory coefficients.