Geometric Properties of Some Generalized Mathieu Power Series inside the Unit Disk

Round 1

Reviewer 1 Report

See the file attached

Comments for author File: Comments.pdf

Author Response

Cover Letter for the revision of
Geometric properties of some generalized Mathieu power series inside the unit disk
submitted to Axioms
To whom it may concern
We have tried to answer all issues raised by the referees and editors. Punctuation and grammar have been corrected at several places, and some additional texts and a conclusion have been added to improve readability.
As requested, we have used the Track Changes package to ensure that the modifications are clearly visible.
One question was whether our results generalize those of [3]. This has been clarified in the text, and our Theorem 2.1 has been slightly extended to make it indeed a generalization of the corresponding theorem of [3].
With best regards
Zivorad Tomovski,

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper has been well-organized and written as well as it seems mathematically correct. However, it needs some revisions from the point of authors and readers to improve the quality of the paper. After these MINOR REVISIONS, I suggest that this paper can be accepted to publish in ”Axioms”.

1. If possible, explain how each of the parameters influences the performance of the proposed approach.

2.  Add the values of alpha and beta of the series in page 2.

3. English is generally good; I think it needs to be polished further and some typos need to be revised. Further punctuation marks should be checked through the paper.

4. Please add the following recently published papers to the reference list to improve the introduction by including the recent development within the frame of geometric function theory.

"Necessary and sufficient conditions for hypergeometric functions to be in a subclass of analytic functions. Afr. Mat. 30(1)(2019), 223-230."

"On subclasses of analytic functions associated with Struve functions. Nonlinear Func. Anal. App. 27(1) (2022), 99-110."

5. It will be more helpful to readers to have some discussions about insight of the main results and outcomes in a conclusion section.

Author Response

Cover Letter
for the revision of
Geometric properties of some generalized Mathieu power series inside the unit disk
submitted to Axioms
To whom it may concern
We have tried to answer all issues raised by the referees and editors. Punctuation and grammar have been corrected at several places, and some additional texts and a conclusion have been added to
improve readability.
As requested, we have used the Track Changes package to ensure that the modifications are clearly visible.
One question was whether our results generalize those of [3]. This has been clarified in the text, and our Theorem 2.1 has been slightly extended to make it indeed a generalization of the corresponding theorem of [3].
With best regards
Zivorad Tomovski,

Author Response File: Author Response.pdf

Reviewer 3 Report

My general impression about the paper is rather positive. It deals with the specific issue of star-like functions, in relation with the class of close-to-convex functions. Does a Mathieu series belong to one of the two classes is the problem that is discussed.

My main concern is related to reference [3], written by two authors, one of them being  author 3 of the present paper. Paper [3] deals with the same subject. Parts of the paper are just repetitions of paper [3] (part of the Introduction, Lemma 1.1, Lemma 1.3).  Furthermore, since results from the paper seem to be closely related to results of [3], explanations should be given before stating Theorems: are the results improvements of [3]? Anyway, independently of any relation with  [3], some introductory sentences should be added before Theorems. The paper is rather technical, and the interest, meaning of Theorems should be explained. Otherwise it becomes a series of computations, formulas.

 

Other comments :

p.1 : "g(z)=z" : why not "\theta=0 and g(z)=f(z)" ?

p.3 : "As the radius of convergence ...". Is not "(a_n) being non-increasing and non-negative" a sufficient and more natural argument?

"convienence", "inherits decrease" : check the correctness of the English

Lemma 1.1 is supposed to be a result from Ozaki, a copy of Lemma 1.1 in [3]. But  the starlike function is -log(1-z) in [3], and becomes  z/(1-z) in the paper. Explanations should be given.

Author Response

Cover Letter
for the revision of
Geometric properties of some generalized Mathieu power series inside the unit disk
submitted to Axioms
To whom it may concern
We have tried to answer all issues raised by the referees and editors. Punctuation and grammar have been corrected at several places, and some additional texts and a conclusion have been added to
improve readability.
As requested, we have used the TrackChanges package to ensure that the modifications are clearly visible.
One question was whether our results generalize those of [3]. This has been clarified in the text, and our Theorem 2.1 has been slightly extended to make it indeed a generalization of the corresponding theorem of [3].
With best regards
Zivorad Tomovski,

Author Response File: Author Response.pdf