Fourth order Hankel determinants for certain subclasses of modified sigmoid-activated analytic functions involving the trigonometric sine function
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Bibliographic record
Abstract
Abstract The aim of this paper is to introduce two new subclasses $\mathcal{R}_{\sin }^{m}(\Im )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mo>sin</mml:mo> <mml:mi>m</mml:mi> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>ℑ</mml:mi> <mml:mo>)</mml:mo> </mml:math> and $\mathcal{R}_{\sin }(\Im )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mo>sin</mml:mo> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>ℑ</mml:mi> <mml:mo>)</mml:mo> </mml:math> of analytic functions by making use of subordination involving the sine function and the modified sigmoid activation function $\Im (v)=\frac{2}{1+e^{-v}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℑ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>v</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> </mml:math> , $v\geq 0$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>v</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:math> in the open unit disc E . Our purpose is to obtain some initial coefficients, Fekete–Szego problems, and upper bounds for the third- and fourth-order Hankel determinants for the functions belonging to these two classes. All the bounds that we will find here are sharp. We also highlight some known consequences of our main results.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| 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) | 0.000 | 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 it