Univalence of Cauchy transforms for infinitesimal arrays

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Let Univ(R){\rm \bf Univ}(\mathbb{R}) be the class of probability measures on R\mathbb{R} with univalent Cauchy transforms, and let IA(⊳){\rm\bf IA}(\rhd) be the class of weak limits of monotone convolutions of infinitesimal arrays of probability measures on R\mathbb{R}. Univalence conjecture.

Univ(R)=IA(⊳).{\rm\bf Univ}(\mathbb{R})={\rm\bf IA}(\rhd).

The preceding theorem establishes the inclusion IA(⊳)⊆Univ(R){\rm\bf IA}(\rhd)\subseteq {\rm\bf Univ}(\mathbb{R}); the converse is posed here and its status is not resolved in the source.

References

Primary source

Uwe Franz, Takahiro Hasebe and Sebastian Schleißinger, “Monotone Increment Processes, Classical Markov Processes, and Loewner Chains”, arXiv:1811.02873 (2020).

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