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Complex inequalities involving sums of holomorphic selfmaps of the unit disk and some experimental conjectures

  • Raymond Mortini
  • Jean-Marc Sac-ÉpéeEmail author
Research
  • 37 Downloads

Abstract

We prove that for every \(p>1\) the set
$$\begin{aligned} D_p=\left\{ c\in {{\mathbb{C}}}\;\big |\; \forall z\in \overline{{{\mathbb{D}}}},\; \left| \frac{1+z}{2}+ c\left( \frac{1-z}{2}\right) ^p\right| \le 1\right\} \end{aligned}$$
is an intersection of closed disks, in particular a closed convex set, and exactly a disk for \(p=2\). It is also shown that
$$\begin{aligned} {{\mathbb{D}}}\cup \{1\}\subseteq \bigcup _{p\ge 2} D_p\subseteq \overline{{{\mathbb{D}}}}. \end{aligned}$$

Keywords

Complex inequalities Holomorphic selfmaps of the disk Experimental conjectures 

Mathematics Subject Classification

30A10 26D05 30J99 

Notes

Acknowledgements

We thank the referee for his/her careful reading of the manuscript.

Supplementary material

40627_2019_37_MOESM1_ESM.avi (16.8 mb)
Supplementary video (AVI 17,176 kb)

References

  1. 1.
    Mortini, R., Carnal, H.: Aufgabe 1350. Elem. Math. 72, 84–85 (2017)Google Scholar
  2. 2.
    Mortini, R., Rhin, G.: Sums of holomorphic selfmaps of the unit disk II. Comput. Methods Funct. Theory 11, 135–142 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Mortini, R., Rupp, R.: Sums of holomorphic selfmaps of the unit disk. Annales Univ. Mariae Curie-Skłodowska 61, 107–115 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Département de Mathématiques et Institut Élie Cartan de Lorraine, UMR 7502Université de LorraineMetzFrance

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