On relations between the classes $\mathcal{S}$ and $\mathcal{U}$

Abstract

Let $\mathcal{A}$ denote the family of all functions $f$ analytic in the unit disk $\ID$ and satisfying the normalization $f(0)=0=f^{\prime }(0)-1$. Let $\mathcal{S}$ denote the subclass of $\mathcal{A}$ consisting of univalent functions in $\ID$. We consider the subclass $\mathcal{U}$ of $\mathcal{S}$ that is defined by the condition that for its members $f$ the condition $$\left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < 1 ~\mbox{ for $z\in \ID$} $$ holds. To theses relations belong striking similarities and on the other hand big differences. We show that some results about $\mathcal{S}$ can be improved for $\mathcal{U}$, while others cannot.

9 pages