Some estimates for angular derivative at the boundary

In this paper, we establish lower estimates for the modulus of the values of $f(z)$ on boundary of unit disc. For the function $f(z)=1+c_1z+c_2z^2+\dots$ defined in the unit disc such that $f(z)\in\mathcal N(\beta)$ assuming the existence of angular limit at the boundary point $b$, the estimations below of the modulus of angular derivative have been obtained at the boundary point $b$ with $f(b)=\beta$. Moreover, Schwarz lemma for class $\mathcal N(\beta)$ is given. The sharpness of these inequalities has been proved.