Conjecture on the starlikeness order of non-linear resolvents

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Let XX be a complex Banach space, let B{\mathbb B} be its open unit ball, and let ff be a holomorphically accretive mapping with an isolated zero at the origin. For each λ>0\lambda>0, let GλG_\lambda be the corresponding resolvent mapping.

Resolvent starlikeness conjecture. Every mapping GλG_\lambda is starlike of order at least 12\frac12. Furthermore, the order of starlikeness of GλG_\lambda tends to 11 as λ→∞\lambda\to\infty.

The conjecture proposes a multidimensional analogue of the known one-dimensional result that resolvents are starlike of order at least 12\frac12, with increasingly strong starlikeness for large resolvent parameter. Its status is not established in the supplied source.

References

Primary source

Mark Elin, “Non-linear resolvents of holomorphically accretive mappings”, arXiv:2406.02758 (2024).

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