On a certain class of starlike functions

Let $\mathcal{S}_u^*$ denote the class of all analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, normalized by $f(0)=f'(0)-1=0$ that satisfies the inequality $\left|zf'(z)/f(z)-1\right|<1$ in $\mathbb{D}$. In the present article, we obtain the sharp estimate of Hankel determinants whose entries are coefficients of $f\in\mathcal{S}_u^*$, logarithmic coefficients of $f\in\mathcal{S}_u^*$ and coefficients of inverse of $f\in\mathcal{S}_u^*$, respectively. We also obtain, the sharp estimate of the successive coefficients for functions in the class $\mathcal{S}_u^*$.