Summary: We present some inequalities for the non-tangential 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 \[ \operatorname{Re} \left( \frac{f^{' }(z)}{\lambda f^{' }(z)+1-\lambda }\right) >\beta ,\quad 0\leq \beta <1,\quad 0\leq \lambda <1, \] we estimate the module of the second non-tangential derivative of \(f(z)\) at the boundary point \( \xi \), by taking into account their first two nonzero Maclaurin coefficients. The sharpness of these estimates is also proved.