In this paper, 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 ${Re}(frac{f^{prime}(z)}{{lambda}f{prime}(z)+1-{lambda}})$ > ${eta}$, $0{leq}{eta}$ < 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.