Hermite–Hadamard Type Inequalities Involving (k-p) Fractional Operator for Various Types of Convex Functions

Round 1

Reviewer 1 Report

The work is well written and organized, the use of symbols is adequate. The results obtained are interesting and novel and may be attractive to a wide circle of readers.

On the penultimate line of page 4, the substitution y=(1-t)a/m+bt is defined, however, it is spelled incorrectly in several lines of the proof below. It must be corrected.

To favor the understanding of the work, the following results should be included, obtained with different operators and notions of convexity.

The Conclusions section should be improved, they could include as an open problem, the use of other notions of convexity and operators.

Author Response

June 20, 2022
Vuk Stojiljkovi´c
Answer to Reviewer 1:
Comment: The work is well written and organized, the use of symbols is adequate. The results
obtained are interesting and novel and may be attractive to a wide circle of readers.
On the penultimate line of page 4, the substitution y = (1 − t)
a
m + bt is defined, however, it is
spelled incorrectly in several lines of the proof below. It must be corrected.
To favor the understanding of the work, the following results should be included, obtained with
different operators and notions of convexity.
The Conclusions section should be improved, they could include as an open problem, the use of
other notions of convexity and operators.
Response: We are very thankful for the reviewer’s excellent suggestions which will greatly
improve the quality of our paper. We have corrected the spelling after the substitution on the page
4. We have provided background on our work and cited the authors whose work is similar to ours.
We have given an open problem in the conclusion.
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, Rajagopalan Ramaswamy , Fahad Sameer Alshammari, Ola Ashour, Mohammed
Lahy Hassan Alghazwani, Stojan Radenovi´c 

Author Response File: Author Response.pdf

Reviewer 2 Report

Kindly check the file.

Comments for author File: Comments.pdf

Author Response

June 20, 2022
Vuk Stojiljkovi´c
Answer to Reviewer 2:
Comment:According to my knowledge and experience, the results presented in the paper are new
and correct. So that I am glad to recommend the paper to your prestigious journal. Before acceptance of the paper, I suggest the following minor changing to make a better presentation of the
paper:
1. Where is the motivation of the paper?! It’s obligatory!
2. Why does this (k - p) Riemann-Liouville fractional integral operator definition have an interest
(if it has of course) for the researchers working in the same domain?
3. It is better to add some nice algebraic properties to this (k - p) RiemannLiouville fractional
integral operator definition.
4. Are these new results sharp and more accurate compared with the others?
5. Please add more lines in the conclusion section.
6. By using a spell checker tool please check all the spelling of the paper.
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.
1)The motivation has been added.
2) We have explained why k − p Riemann Liouville fractional operator is of interest.
3)We have added algebraic properties of the k − p Riemann-Liouville fractional operator.
4)Sorry to say so, but we are not able to provide an answer to this question.
5)We have added an open problem in the conclusion section.
6)We have checked all the spelling in the paper.
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, Rajagopalan Ramaswamy , Fahad Sameer Alshammari, Ola Ashour, Mohammed
Lahy Hassan Alghazwani, Stojan Radenovi´c

Author Response File: Author Response.pdf

Reviewer 3 Report

In the paper under review authors establish various interesting fractional convex inequalities of the Hermite-Hadamard type involving Involving (k-p) Fractional Operators. There are some minor remarks regarding style and presentation. In the statement of Definition 4 it is not clear what actually is defined there and it should be written with more care. In the Remark 1 it should be stated which definition you are referring to. Statement of Theorem 1 should be reformulated, for example: ...one of the cases 1. ... or 2. ... the following inequality hold... References should be written consistently.

Main concern here is the already published paper:
Soubhagya Kumar Sahoo, Hijaz Ahmad, Muhammad Tariq, Bibhakar Kodamasingh, Hassen Aydi and Manuel De la Sen:
Hermite–Hadamard Type Inequalities Involving k-Fractional Operator for (h,m)-Convex Functions,
Symmetry 2021, 13, 1686. https://doi.org/10.3390/sym13091686
in which similar problem was discussed. In my opinion this paper should be included in the references and authors should make clear comparison of their results with the results from the above mentioned paper.

 

Author Response

June 20, 2022
Vuk Stojiljkovi´c
Answer to Reviewer 3:
Comment:
In the paper under review authors establish various interesting fractional convex inequalities of
the Hermite-Hadamard type involving Involving (k-p) Fractional Operators. There are some minor
remarks regarding style and presentation. In the statement of Definition 4 it is not clear what
actually is defined there and it should be written with more care. In the Remark 1 it should be
stated which definition you are referring to. Statement of Theorem 1 should be reformulated, for
example: ...one of the cases 1. ... or 2. ... the following inequality hold... References should be
written consistently.
Main concern here is the already published paper: Soubhagya Kumar Sahoo, Hijaz Ahmad,
Muhammad Tariq, Bibhakar Kodamasingh, Hassen Aydi and Manuel De la Sen: Hermite–Hadamard
Type Inequalities Involving k-Fractional Operator for (h,m)-Convex Functions, Symmetry 2021, 13,
1686. https://doi.org/10.3390/sym13091686 in which similar problem was discussed. In my opinion
this paper should be included in the references and authors should make clear comparison of their
results with the results from the above mentioned paper.
Response: We thank the reviewer for his/her helpful comments. We have rewritten the Definition 4. We have improved the Remark 1. We are very thankful to the great reviewers advice to
cite the paper he/she mentioned. We have mentioned it in the section before the main results and
gave it appreciation.
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, Rajagopalan Ramaswamy , Fahad Sameer Alshammari, Ola Ashour, Mohammed
Lahy Hassan Alghazwani, Stojan Radenovi´c 

Author Response File: Author Response.pdf

Round 2

Reviewer 3 Report

Check the references (and the rest of the paper) for spelling errors (capital letters, spaces, commas, etc.)

Author Response

June 24, 2022
Vuk Stojiljkovi´c
Answer to Reviewer 1:
Comment: Check the references (and the rest of the paper) for spelling errors (capital letters,
spaces, commas, etc.)
Answer: We have checked everything in great detail. Once again we thank the referee for
his/her 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, Rajagopalan Ramaswamy , Fahad Sameer Alshammari, Ola Ashour, Mohammed
Lahy Hassan Alghazwani, Stojan Radenovi´c 

Author Response File: Author Response.pdf