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Kraetzer’s universal integral-means spectrum formula

For normalized univalent disk maps, determine the p=−1 universal integral-means spectrum. The counterexample target is a uniform M_−1[f′](r)≤C(1−r)^(−1/4+ε), ε>0, implying B_b(−1)<1/4 for the bounded-image spectrum.

Why it matters and what counts as progress

Why it matters: A strict bound below one quarter would contradict the predicted spectrum formula at this exponent.

What counts as progress: A strict improvement below the predicted p=−1 spectrum value, or a corrected universal integral-means formula.

How the problem developed

  1. 1996 · Conjecture [1]

    P. Kraetzer Proposed the quadratic formula for the universal integral-means spectrum on |p|≤2, based on computer experiments.

  2. 24 Sept 2026 · Counterexample to prediction [2]

    OpenAI research team Proved B_b(−1)<1/4, contradicting the predicted value at p=−1.

  3. 16 Oct 2026 · Stronger bound [3]

    Ilia Binder, Oleg Ivrii, István Prause and Stanislav Smirnov Reported B_b(t)≤t²/4 on [−2,2], with strict inequality in the interior except at zero, while noting that details of the strictness argument are deferred.

AI contribution

Sources

  1. P. Kraetzer, Experimental bounds for the universal integral means spectrum of conformal maps
  2. OpenAI, Kraetzer's conjecture on the universal integral means spectrum
  3. Ilia Binder et al., Brennan’s Conjecture and the Universal Integral Means Spectrum
  4. A strict inverse-first-power bound for univalent functions
  5. Lean proof of OpenAI reports a strict bound that refutes Kraetzer’s spectrum formula
  6. OpenAI mathematics release and provenance
  7. OpenAI, Brennan’s conjecture and sharp inverse-square integral means
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