On a certain class of starlike functions

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