Review Manuscript ID: symmetry-1002950 Type of manuscript: Article Title: COEFFICIENT BOUNDS FOR CERTAIN CLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH FABER POLYNOMIAL Authors: Adel A. Attiya, Abdel Moneim Lashin, EKRAM E. ALI, Praveen Agarwal* Submitted to the section: Mathematics and Symmetry, Advanced Calculus in Problems with Symmetry Date: 23 January 2021

The article describes Faber polynomials. Here are some remarks.

• The introduction should include a general overview of the problem, not suddenly jump into a bunch of formulas. Keep in mind that this journal is not a (pure) mathematics journal per se, but also include applications in the field outside mathematics.

• The definition of Faber polynomials should be provided.

• The references on Faber polynomials should be given and cited accordingly, e.g.,

• Curtiss, J. H. (1971), "Faber Polynomials and the Faber Series", The American Mathematical Monthly, Mathematical Association of America, 78 (6): 577–596, doi:10.2307/2316567, ISSN 0002-9890, JSTOR 2316567
• Faber, Georg (1903), "Über polynomische Entwickelungen" (PDF), Mathematische Annalen, Springer Berlin / Heidelberg, 57: 389–408, doi:10.1007/BF01444293, ISSN 0025-5831
• Faber, G. (1919), "Über Tschebyscheffsche Polynome.", Journal für die reine und angewandte Mathematik (in German), 150: 79–106, ISSN 0075-4102, JFM 47.0315.01
• Grunsky, Helmut (1939), "Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen", Mathematische Zeitschrift, 45 (1): 29–61, doi:10.1007/BF01580272, ISSN 0025-5874
• Schur, Issai (1945), "On Faber polynomials", American Journal of Mathematics, 67: 33–41, doi:10.2307/2371913, ISSN 0002-9327, JSTOR 2371913, MR 0011740
• Suetin, P. K. (1998) [1984], Series of Faber polynomials, Analytical Methods and Special Functions, 1, New York: Gordon and Breach Science Publishers, ISBN 978-90-5699-058-9, MR 1676281

• Suetin, P. K. (2001) [1994], "Faber polynomials", Encyclopedia of Mathematics, EMS Press

• Page 3: the definition of and (references/citations on) the Schwarz function should be given and provided.

• Page 3 typo: a domain

• Page 4: the abbreviation SD(\alpha,\beta) and KD(\alpha,\beta) should be explained.

• Page 4: the symbol for the real part is not consistent: sometimes using the fraktur font "R", but other times using the regular font "Re". Please make it uniform.

• Page 5: what is univalent?

• Page 5: make the citation compact, i.e., [3-5, 9-10, ..., 36-40], etc.

• Page 5: what is fraktur R_\sigma?

• Page 7: what does yield from using the Faber polynomial?

• Theorem 1: the summation starts from n = 2, but the result is valid starting from n \geq 3. Which one is correct? This is confusing. Please clarify.

• Page 9: adjust the size of the bracket.

• Definition 7: adjust the space between "if" and "the".

• Note that the readers of this journal come from various backgrounds, including Applied Mathematics. Hence, a few words on applications would be useful to aid general readers to understand what the topic is about.