Bibliographic record
Abstract
Abstract Complex analysis of a single variable is a type of mathematical analysis that observe properties for holomorphic functions. Over decades of development, complex analysis nowadays has played a more and more important role in algebraic geometry, fluid dynamics, quantum mechanics, and so on. This paper is to elaborate the Brannan’s coefficient conjecture, which was first proposed by D.A Brannan, J.G. Clunie, and W.E Kirwam in their paper on the coefficient problem for functions of bounded boundary rotation in 1973. The purpose of this paper is to give a detailed review and exploration of complicated but elegant proofs by D. Aharonov and S. Friedland. This paper mainly focuses on proving the inequality <?CDATA $\frac{{(1+\alpha x)}^{n}}{{(1-\text{x})}^{\text{p}}}\le \frac{{(1+x)}^{n}}{{(1-\text{x})}^{\text{p}}}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>α</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> for p > 0 | α | = 1 within condition n ≥ 1, which was derived from Brannan’s coefficient conjecture. The proof of the inequality is done by dividing the techniques of complex numbers, trigonometric functions, binomial expansions, and other mathematical formulas into three different cases.
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How this classification was reachedexpand
Full frame distilled prediction
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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".