On a special class of Schwartz functions

Abstract

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 $|{\omega }^{\prime }(z)|\le 1$ for all $z\in \mathbb{D}$. For these functions we give estimates (sometimes sharp) for the following moduli: $|c_3-c_1{c}_2|$, $|c_1{c}_3-c_2^2|$, and $|c_4-c_2^2|$.