On a special class of Schwartz functions

In this paper we study functions $\omega (z)=c_{1}z+c_2{z}^2+c_3{z}^3+\cdots$ analytic in the open unit disk $\mathbb{D}$ and such that $\mathrm{\mid }{\omega }^{\mathrm{\prime }}(z)\mathrm{\mid }\le 1$ for all $z\in \mathbb{D}$. For these functions we give estimates (sometimes sharp) for the following moduli: $\mathrm{\mid }{c}_3-c_1{c}_2\mathrm{\mid }$, $\mathrm{\mid }{c}_1{c}_3-c_2^2\mathrm{\mid }$, and $\mathrm{\mid }{c}_4-c_2^2\mathrm{\mid }$.