Let ${\mathcal{SR}}^*$ be the class of starlike functions with real coefficients, i.e., the class of analytic functions $f$ which satisfy the condition $f(0)=0=f'(0)-1$, Re{z f'(z) / f (z)} > 0, for $z\in\mathbb{D}:=\{z\in\mathbb{C}:|z|<1 \}$ and $a_n:=f^{(n)}(0)/n!$ is real for all $n\in\mathbb{N}$. In the present paper, the sharp estimates of the third Hankel determinant $H_{3,1}$ over the class ${\mathcal{SR}}^*$ are computed.

Let BT4l denotes a subclass of bounded turning functions connected with a four-leaf-type domain. The goal of the study is to probe into coefficients of |b6|, |b7|, |b8|, the bounds of the logarithmic coefficients and the third-order determinants of H3,1, H3,2, H3,3 for functions in this class.

In this work, some numerical radius inequalities based on the recent Dragomir extension of Furuta's inequality are obtained. Some particular cases are also provided.