In this paper, we discuss different versions of the boundary Schwarz lemma and Hankel determinant for $\mathcal{K}\left( \alpha \right) $ class. Also, for the function $f(z)=z+c_{2}z^{2}+c_{3}z^{3}+\cdots$ defined in the unit disc such that $f\in \mathcal{K(\alpha )}$, we estimate a modulus of the angular derivative of $f(z)$ function at the boundary point $z_{0}$ with $f(z_{0})=\frac{z_{0}}{1+\alpha }$ and $f^{\prime }(z_{0})=\frac{1}{1+\alpha }$. That is, we shall give an estimate below $\left\vert f^{\prime \prime }(z_{0})\right\vert $ according to the first nonzero Taylor coefficient of about two zeros, namely $z=0$ and $z_{1}\neq 0$. The sharpness of this inequality is also proved.