Applications of the q-Derivative Operator to New Families of Bi-Univalent Functions Related to the Legendre Polynomials

Round 1

Reviewer 1 Report

I recommend the publication of the manuscript after the authors have done some revision to the paper.

The authors should consider the following comments

Comments for author File: Comments.docx

Author Response

1. We add other examples of the structures studied in Section 1.
2. We add the statements before Corollaries 1 and 2 which explain the special cases.
3. The Acknowledgement is added at the end of the text.

Reviewer 2 Report

Paper: Applications of the q-Derivative Operator to New Families of

Bi-Univalent Functions Related to the Legendre Polynomials

The authors introduce new subclasses of q-starlike functions and estimate coefficient and inequalities of these classes.

It is clearly, as authors mention, an extension of previous works obtained by them.

The Introduction section is presented as a tool section, where they are presented properly, focusing the main structures (notations, definition, and lemmas) needed to support the main results.

Due this type of work the authors toke the decision to ignore the presence of a methodology section, going directly from introduction to the main results. They keep it simple and direct this way, since all material needed to stablish the proof of each theorem is there, calling the proper references when needed.

Nevertheless, my recommendation goes for the need of split the actual introduction in two sections. Presenting, perhaps, a summarized version of some of the nice introductions that we can find on the previous works of the same authors and discussing briefly the differences founded in other papers about this subject. In the second part keeping the tools, presented in the present paper version, and complete with some more information about its importance and future use in the methodological way of obtaining the theorem’s proof.

The main results section is fine as it is.

In the conclusions section I would remove the offensive words as “amateurish-type publications” and “falsely-claimed” by others that can express the same authors opinion about other publications without being offensive. Perhaps, this paragraph should be used as a door to structure a proper methodological section missing on the paper, showing that the authors way, of organizing ideas and concepts, is better and results in the elimination of the so called “falsely-claimed generalizations”.

Author Response

According to your kind suggestions, we have revised our paper.
1. We split the Introduction in two sections in view of your suggestion.
2. The Acknowledgement is added at the end of the text.

Reviewer 3 Report

My recommendation for authors is to provide more of an overview of the publications in the introduction.

Comments for author File: Comments.pdf

Author Response

1. According to your kind suggestions, we have revised the Introduction of the paper.
2. The Acknowledgement is added at the end of the text.