Coefficient Estimates and Fekete–Szegö Functional Inequalities for a Certain Subclass of Analytic and Bi-Univalent Functions

Round 1

Reviewer 1 Report

In this paper, authors introduced a new class of bi-univalent functions defined on symmetric domain using Gegenbauer polynomials. Authors also derived the estimates of the Taylor-Maclaurin coefficients and Fekete-Szego functional problems for functions belonging to the new subclasses. 

Extensive proof reading and spell check is required for the paper. Authors made some typos: for example Theorm 2.2 after line 54 and Theorm 2.3 in the line 58, which is supposed to be Theorem 2.2 and Theorem 2.3 respectively. In corollary 3.1, there is a latex error, "Where" is supposed is supposed to be in the last line not in the second last line. 

Authors have given definitions and preliminaries required for the paper and literature review is okay. Clearly it is easy to follow their work from the very beginning. I would like to appreciate their hard work. 

The coefficients estimates given by the theorem 2.2 is very interesting to me. The proof is given thoroughly and is very satisfactory result. However, the proof of the Fekete-Szego inequality on the line 59 ( i.e theorem 2.3) is not in detail and not obvious to me. I would like to see more details of the proof. 

In both theorems 2.2, and 2.3 and in all corollaries, authors have derived and studied coefficients estimates and consequently they deduced a large algebraic expressions as an estimates. I am curious if those estimates have any mathematical significance. If so, it would be great if authors indicated that significance in general. 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

see the attachment.

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors prove some coefficient estimates for bi-univalent functions.

In Definition 2.1 they introduce a class of such functions, depending on several parameters, and Theorems 2.2 and 2.3 (the main results) then give estimates for the first two coefficients for functions belonging this class. The estimates are relatively involved (complicated), but the paper as a whole is well-written and clear. It is difficult for a non-specialist to estimate the value of the result and to check the proofs in detail, but the paper looks sound in a general sense and I feel confidence in it. Therefore I am willing to recommend the it for publication.

Author Response

Thank you for your positive feedback.