Geodesic
Another look at the Landau theorem
Zhenghua Xu1
1 Hefei University of Technology, School of Mathematics
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 581-585.

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  Let $f$ be the analytic self-map of the open unit disk $\mathbf{D}$ in complex plane with $f(0)=0$ and $f'(0)\neq 0$. The classical Landau theorem shows that the image $f(\mathbf{D})$ contains a schlicht disk. In this note, it is proved that the injectivity of $f$ in the Landau theorem can be strengthened to be starlike. In particular, it provides a way to construct starlike functions from bounded analytic functions. Furthermore, we obtain a new version of the Landau theorem for vector-valued analytic functions from the perspective of modulus functions.
Keywords: Vector-valued analytic function, Landau theorem, starlike function

Zhenghua Xu 1

1 Hefei University of Technology, School of Mathematics 
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Zhenghua Xu. Another look at the Landau theorem. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 581-585. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a34/