An Extended Hilbert-Type Inequality with Two Internal Variables Involving One Partial Sums
Round 1
Reviewer 1 Report
Referee Report
Manuscript ID: axioms-2552187
Title: A New Extended Hardy-Hilbert’s Inequality with the Internal Variables Involving One Partial Sums
Authors: Aizhen Wang, Bicheng Yang
Abstract Review:
The abstract provides a concise overview of the paper's content. It effectively conveys that the manuscript presents a new extended version of the Hardy-Hilbert’s inequality with power functions as internal variables, specifically involving one partial sum. It also highlights the use of weight coefficients, real analysis techniques, and the mid-value theorem in deriving the inequality. The claim of the refinement of a previously published inequality and the identification of equivalent statements for optimal constant factors lend significance to the work. Mention of equivalent forms, operator expressions, and specific inequalities derived from this work further emphasize its applicability.
Content Review:
The paper delves into a topic of interest and importance in the field of mathematical inequalities. The authors skillfully employ established techniques, such as weight coefficients and real analysis methods, to derive a novel extension of the Hardy-Hilbert’s inequality. This extension, involving power functions as internal variables tied to one partial sum, demonstrates a refined form of a previously published inequality.
The mathematical rigor exhibited throughout the manuscript is commendable. The application of the mid-value theorem adds depth to the analysis and enhances the novelty of the findings. The presentation of equivalent statements for the best possible constant factor involving various parameters further enriches the paper's contributions. Additionally, the exploration of equivalent forms, operator expressions, and specific inequalities as applications showcases the versatility of the derived results.
The organization of the manuscript is well-structured, allowing readers to follow the logical progression of ideas. However, a minor point of improvement could be the clarity of certain mathematical derivations. A few steps could benefit from additional explanation to aid readers less familiar with the specific techniques employed.
Recommendation:
Based on the thorough review of the manuscript, I recommend that it be accepted for publication in its current form. The paper contributes to the field of mathematical inequalities by presenting a novel extension of the Hardy-Hilbert’s inequality, incorporating well-established techniques and results. The authors' clear articulation of the mathematical derivations, combined with the paper's practical implications, renders it a valuable addition to the Axioms journal.
The manuscript aligns well with the journal's scope and aims to advance mathematical knowledge. It is anticipated that this work will be of interest to researchers and practitioners working in the field of mathematical inequalities and related areas. With minor adjustments to enhance the clarity of certain derivations, this paper has the potential to make a positive impact on the academic community.
Overall, I recommend accepting the manuscript in its current form.
Just minor revisions in english
Author Response
Please see the attached file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper is scientifically correct.
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Comments for author File: Comments.pdf
Author Response
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Author Response File: Author Response.pdf
Reviewer 4 Report
The authors write,
In this article, following the way of [28], by means of the weight coefficients and the idea of introduced parameters, applying Euler-Maclaurin summation formula, Abel’s summation by parts formula, and the mid-value theorem, a new extended Hardy-Hilbert’s inequality with the power function as the internal variables involving one partial sums is given, which is a refinement of the inequality in [27]. The equivalent statements of the best possible constant factor related to several parameters are provided. The equivalent forms, the operator expressions and some particular inequalities are obtained as applications.?
But I can not see in the References the papers from (26) to (29) ?
Therfore ,I can not estimate ,
Literature Review* Methodology & Procedures The novelty of the results , Clarity of Expression etc.,
Also, another ones in the References,
(1),(4), (34) are books but written with small letters ,the reference of paper (24) is not full.
Author Response
Please see the attached file.
Author Response File: Author Response.pdf
Round 2
Reviewer 4 Report
the revised version of the paper is already publishable.
