Communications of the
Korean Mathematical Society CKMS

In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function $f(z)$ holomorphic in the unit disc and $ f(0)=0 $, $f^{\prime }(0)=1$ such that $\Re f^{\prime }(z)>\frac{1-\alpha }{2 }$, $-1<\alpha <1$, we estimate a modulus of the second non-tangential derivative of $f(z)$ function at the boundary point $z_{0}$ with $\Re f^{\prime }(z_{0})=\frac{1-\alpha }{2}$, by taking into account their first nonzero two Maclaurin coefficients. Also, 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 these inequalities is also proved.

Keywords: Schwarz lemma on the boundary, holomorphic function, second non-tangential derivative, critical points

MSC numbers: Primary 30C80, 32A10, 58K05