Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus

Round 1

Reviewer 1 Report

In the paper, sharper upper and lower bounds are obtained in terms of the polynomials.Based on four theorems and related corollaries, the estimation formulas of a kind of integral and two kinds of fractional integral are given.The application of inequality studied in this paper should be listed several, to illustrate the significance of the study.

Author Response

 We are very thankful for the reviewer’s excellent suggestions which will greatly
improve the quality of our paper. We have added a Corollary in Theorem 1 and in Theorem 2 and
have written how the Corollary can be used in other Theorems. Kindly see Corollary 2,Corollary 4.
Waiting eagerly to hear from you a positive response in this regard.
With best regards.
Sincerely yours
Vuk Stojiljkovi´c
On behalf of the authors:
Vuk Stojiljkovi´c, Slobodan Radojevi´c, Ey¨up C¸ etin, Vesna Seˇsum Cavi´c, Stojan Radenovi´c 

Author Response File: Author Response.pdf

Reviewer 2 Report

See the attached file.

Comments for author File: Comments.pdf

Author Response

: Thank you very much for giving us your valuable comments. We especially appreciate the reviewer’s labour for evaluating our paper. We will continue to work hard. Our answer is
as follows.
i) We have written the name of l hopital correctly.
ii) We have changed the word inclusion as said.
iii)We have rewritten the conditions in the Corollary and are very thankful for the remarks, as it
greatly improved the readability of the Corollary.
Waiting eagerly to hear from you a positive response in this regard.
With best regards.
Sincerely yours
Vuk Stojiljkovi´c
On behalf of the authors:
Vuk Stojiljkovi´c, Slobodan Radojevi´c, Ey¨up C¸ etin, Vesna Seˇsum Cavi´c, Stojan Radenovi´

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and suggestions are in the pdf file. 

Comments for author File: Comments.pdf

Author Response

Thank the reviewer for his/her helpful comments. We have added the definition of
the F1 functions at the beginning of the paper. We also have discussed the convergence issue in the
first Corollary of the paper. We thank the referee very much for his/her helpful comments.
Waiting eagerly to hear from you a positive response in this regard.
With best regards.
Sincerely yours
Vuk Stojiljkovi´c
On behalf of the authors:
Vuk Stojiljkovi´c, Slobodan Radojevi´c, Ey¨up C¸ etin, Vesna Seˇsum Cavi´c, Stojan Radenovi´c

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Dear authors, 

The part including the applications to fractional calculus have been substantially improved. 

But the proof of Theorem 1 and the statements and proofs of Theorems 2, 4 and 3 can be improved, according to the first review report sent by me.  In the proof of Theorem 1, the part proving the nonnegativity of the derivative of f is too intricate and not so clear. Theorems 2,3,4 can be generalized easily-you should  provide some references with similar inequalities, less general. 

Regarding the style, please avoid to begin so many sentences with "Which". 

What do you mean by "The functions on which we apply the Riemann-Liovuille fractional integral are well  defined in terms of the integral formula."? Please correct to Liouville. It would be recommended to specify the minimal regularity of functions for which the Riemann-Liouville integral transform is applicable, i. e. the functions are assumed to be locally  integrable. 

Regards

Author Response

June 12, 2022
Vuk Stojiljkovi´c
Answer to Reviewer :
Comment: But the proof of Theorem 1 and the statements and proofs of Theorems 2, 4 and 3 can
be improved, according to the first review report sent by me. In the proof of Theorem 1, the part
proving the nonnegativity of the derivative of f is too intricate and not so clear. Theorems 2,3,4
can be generalized easily-you should provide some references with similar inequalities, less general.
Regarding the style, please avoid to begin so many sentences with ”Which”.
What do you mean by ”The functions on which we apply the Riemann-Liovuille fractional
integral are well defined in terms of the integral formula.”? Please correct to Liouville. It would
be recommended to specify the minimal regularity of functions for which the Riemann-Liouville
integral transform is applicable, i. e. the functions are assumed to be locally integrable.
Response: We are very thankful for the reviewer’s excellent suggestions which will greatly
improve the quality of our paper.
i) We have added the condition of local-integrability in the definition of the Riemann-Liouville
Fractional integral. We also have corrected the Liouville.
ii) We have shortened the proof of Theorem 1, which should make it clear. We have shortened the
proof in Theorem 3 using your suggestions.
iii) We have put a comment in the conclusion section regarding the generalization of the method,
using Taylor expansion.
We are very thankful for your useful suggestions which make the paper much better and more
reader-friendly.
Waiting eagerly to hear from you a positive response in this regard.
With best regards.
Sincerely yours
Vuk Stojiljkovi´c
On behalf of the authors:
Vuk Stojiljkovi´c, Slobodan Radojevi´c, Ey¨up C¸ etin, Vesna Seˇsum Cavi´c, Stojan Radenovi´c

Author Response File: Author Response.pdf