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arxiv: 1907.07827 · v1 · pith:D4XW4YRTnew · submitted 2019-07-18 · 🧮 math.CA · cs.DM

A study of multivalent q-starlike functions connected with circular domain

Lei Shi , Qaiser Khan , Gautam Srivastava , Jin-Lin Liu , Muhammad Arif This is my paper

Pith reviewed 2026-05-24 19:46 UTC · model grok-4.3

classification 🧮 math.CA cs.DM
keywords multivalent functionsq-starlike functionscircular domainconvolutioncoefficient estimatesFekete-Szego inequalitiesBernardi integral operatorq-calculus
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The pith

A new class of multivalent q-starlike functions tied to circular domains yields coefficient estimates and Fekete-Szego inequalities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new subfamily of analytic multivalent q-starlike functions connected to a circular domain and studies its basic properties. It derives results on convolution, sufficiency criteria for class membership, bounds on Taylor coefficients, and Fekete-Szego type inequalities. The authors also introduce a q-extension of the Bernardi integral operator and examine its action on the new class. These steps extend classical techniques from geometric function theory into a q-calculus setting for multivalent mappings.

Core claim

The authors introduce the class of multivalent q-starlike functions associated with the circular domain and prove that functions in this class satisfy convolution closure properties under stated conditions, admit explicit sufficiency criteria, obey coefficient estimates obtained via subordination, and fulfill Fekete-Szego inequalities; they further define the q-Bernardi integral operator and establish its basic mapping properties for the class.

What carries the argument

The newly defined subfamily of multivalent q-starlike functions linked to the circular domain via a subordination condition, which enables direct application of standard coefficient and inequality techniques.

Load-bearing premise

The newly defined subfamily of multivalent q-starlike functions satisfies the analyticity and mapping conditions in the circular domain that permit direct application of standard subordination and coefficient techniques without additional restrictions or counterexamples.

What would settle it

A concrete analytic multivalent function that satisfies the defining subordination condition for the new class yet violates one of the stated coefficient bounds or Fekete-Szego inequalities would disprove the claims.

read the original abstract

In the present article, our aim is to examine some useful problems including the convolution problem, sufficiency criteria, coefficient estimates and Fekete-Szego type inequalities for a new subfamily of analytic and multivalent functions associated with circular domain. In addition, we also define and study a Bernardi integral operator in its $q$-extension for multivalent functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript defines a new subfamily of analytic multivalent q-starlike functions associated with a circular domain and investigates convolution problems, sufficiency criteria, coefficient estimates, Fekete-Szegő inequalities, and a q-analogue of the Bernardi integral operator.

Significance. If the class definition is consistent with the stated mapping properties and the derivations hold, the work provides incremental extensions of q-geometric function theory to the multivalent case in a circular domain, along with associated bounds and an integral operator. Such results can serve as references for further studies in q-calculus applications to analytic functions, though the contribution is primarily technical rather than foundational.

minor comments (2)
  1. [Abstract] Abstract: the description of the circular domain and the precise role of the parameter q remain high-level; a brief indication of the defining subordination or analytic condition would improve clarity for readers.
  2. The manuscript would benefit from an explicit statement of the radius of the circular domain and any restrictions on q to ensure the functions remain analytic and multivalent in the unit disk.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing the manuscript. The report summarizes the work but provides no specific major comments to address point by point. We note the uncertain recommendation and the characterization of the contribution as incremental; the paper aims to extend q-geometric function theory to the multivalent setting with a circular domain, including new results on convolutions, coefficients, Fekete-Szegő inequalities, and the q-Bernardi operator.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new class of multivalent q-starlike functions tied to a circular domain and applies standard subordination and coefficient techniques to obtain convolution properties, sufficiency criteria, bounds, and a q-Bernardi operator. No quoted equations or derivation steps reduce any claimed prediction, bound, or result to the class definition by construction. No self-citation chains, imported uniqueness theorems, or ansatzes smuggled via prior work are indicated in the provided text. The work is a conventional incremental extension in geometric function theory whose central claims retain independent mathematical content beyond the initial definition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; insufficient detail to enumerate specific free parameters, axioms, or invented entities beyond the general setting of q-calculus and multivalent analytic functions.

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discussion (0)

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Reference graph

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