General-coefficient C*-algebraic Bieberbach bounds

Let $A$ be a unital C*-algebra. Suppose the normalized power series

F(u)=u+\sum_{n\ge2}a_nu^n

converges in the norm of $A$ and is injective on the norm-open unit ball $\{u:\|u\|<1\}$ . Must

a_na_n^*\le n^2 1_A,\qquad a_n^*a_n\le n^2 1_A\quad(n\ge2)?

Does the assertion hold if the formal derivative series is additionally required to converge in norm?

These are the general-coefficient Conjectures 1.2 and 1.4 of Krishna's 2022 preprint. The linked paper refutes both with one entire, globally injective series over $C([0,1],\mathbb C)$ . The coefficients are not required to be invertible in these two conjectures. The source's distinct invertible-coefficient variants, polynomial variants, Robertson and Lebedev–Milin assertions are not claimed resolved. The scalar Bieberbach theorem is untouched.

References

K. Mahesh Krishna, C*-algebraic Bieberbach, Robertson, Lebedev–Milin, Zalcman, Krzyz and Corona Conjectures, Preprints (2022), v1, Conjectures 1.2 and 1.4. https://doi.org/10.20944/preprints202208.0140.v1 Author-deposited source: https://doi.org/10.5281/zenodo.6917925 Answering AI-assisted preprint: https://mathdb.com/abs/2610.00792 Lean proof package: https://doi.org/10.5281/zenodo.23718291

Posted by MJ KangPosted 3 days ago · edited 3 days ago

Progress summary

Refreshed 3 days ago

Open · solution reported

A new preprint claims a counterexample disproves both general-coefficient bounds, while related variants remain open.

Krishna posed these two conjectures in 2022: whether injectivity of a normalized power series on the unit ball of a unital $C^*$ -algebra forces the Bieberbach-type coefficient bounds, with or without norm convergence of the formal derivative.

Claimed October 2026 counterexample

Myeongjin Kang's preprint claims that a single entire, globally injective series over $C([0,1],\mathbb C)$ violates the coefficient bound, thereby refuting both Conjectures 1.2 and 1.4. The claim does not address the invertible-coefficient, polynomial, Robertson, or Lebedev–Milin variants, and leaves the scalar Bieberbach theorem untouched.

Current status (as of October 2026): Both general-coefficient conjectures are claimed false by one counterexample, but the preprint's mathematical resolution remains unverified; the stated variants remain open.

Sources

A normalized norm-entire power series on C([0,1],C) is globally injective and has an everywhere norm-convergent formal derivative series, but its second coefficient violates the general C*-Bieberbach order bound. A continuous exponential period anchored at zero proves global injectivity without assuming univalent scalar fibers. Lean proves the full source Conjecture 1.2/1.4 counterexample as one common outcome.See full solutionHide full solution

AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review. The accompanying Lean certificate covers the precise scoped main statement; inspect the formal endpoint, assumptions and provenance in the supporting package. Prior-art investigations are documented bounded searches, not worldwide-priority certification. Original prose is offered under CC BY 4.0 and original code/Lean under MIT; third-party rights and existing notices are preserved. The public supporting ZIP contains original proof sources and verification evidence, with bibliographic links and hashes instead of redistributed third-party source manuscripts.

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Full paper: An injective C*-power series violating the Bieberbach coefficient bound.

Lean proof sources and pinned verification evidence: Zenodo supporting package. MathDB has not automatically verified the attached Lean certificate.