On the Koebe Quarter Theorem for Certain Polynomials of Even Degree
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Bibliographic record
Abstract
Abstract We continue research on problems similar to the Koebe Quarter Theorem for close-to-convex polynomials with all zeros of derivative in $$\mathbb T:=\{z\in \mathbb C:|z|=1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mi>z</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo>:</mml:mo> <mml:mo>|</mml:mo> <mml:mi>z</mml:mi> <mml:mo>|</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> . We found the minimal disc containing all images of $$\mathbb D:=\{z\in \mathbb C: |z|<1\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mi>z</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo>:</mml:mo> <mml:mo>|</mml:mo> <mml:mi>z</mml:mi> <mml:mo>|</mml:mo> <mml:mo><</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> and the maximal disc contained in all images of $$\mathbb D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> through polynomials of degree 6. Moreover, we determine the extremal functions for both problems. Furthermore, we state the conjecture concerning polynomials of higher even degrees.
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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.010 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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