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arXiv · 1604.07733

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

Milutin Obradović·Saminathan Ponnusamy·Karl-Joachim Wirths
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Abstract

Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the unit disk $\ID$ and satisfying the normalization $f(0)=0= f'(0)-1$. 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.

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Milutin Obradović, Saminathan Ponnusamy, Karl-Joachim Wirths. 2016-04-26. On relations between the classes $\mathcal S$ and $\mathcal U$. https://arxiv.org/abs/1604.07733

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