Communications of the
Korean Mathematical Society CKMS

In this paper, we present some inequalities for the non-tan\-gential derivative of $f(z)$. For the function $ f(z)=z+b_{p+1}z^{p+1}+b_{p+2}z^{p+2}+\cdots$ defined in the unit disc, with $ \Re \left( \frac{f^{\prime }(z)}{\lambda f^{\prime }(z)+1-\lambda }\right) >\beta $, $0\leq \beta <1$, $0\leq \lambda <1$, we estimate a module of a second non-tangential derivative of $f(z)$ function at the boundary point $ \xi $, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.

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

MSC numbers: 30C80, 32A10, 58K05