Third Hankel Determinant and Zalcman Functional for Sakaguchi Type Starlike Functions Involving q-Derivative Operator Related with Sine Function
Bibliographic record
Abstract
The purpose of this paper is to consider coefficient estimates for q-starlike function with respect to symmetric points associated with sine function \(\mathcal{SS}^*_ q(1+sin(z))\) consisting of analytic functions \(f\) normalized by \(f(0)=f'(0)-1=0\) in the open unit disk \(\mathcal{U}_d=\{ z:z\in \mathbb{C}\quad \text{and}\quad \left\vert z\right\vert <1\}\) satisfying the condition \(\dfrac{2[zD_qf(z)]}{f(z)-f(-z)}\prec{1+sin(z)}=\psi(z)\), for all \(z\in\mathcal{U}_d\) to derive certain coefficient estimates \(b_2,b_3\) etc and Fekete-Szeg\"{o} inequality for \(f\in\mathcal{SS}^*_q(1+sin(z)).\) Further to investigate the possible upper bound of third order Hankel determinant and also the Zalcman functional for \(f\in\mathcal{SS}^*_q(1+sin(z))\).
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 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) | 0.000 | 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".