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Peer-Review Record

Novel Approaches for Differentiable Convex Functions via the Proportional Caputo-Hybrid Operators

Fractal Fract. 2022, 6(5), 258; https://doi.org/10.3390/fractalfract6050258
by Mustafa Gürbüz 1, Ahmet Ocak Akdemir 2,* and Mustafa Ali Dokuyucu 2
Reviewer 1:
Reviewer 2: Anonymous
Fractal Fract. 2022, 6(5), 258; https://doi.org/10.3390/fractalfract6050258
Submission received: 19 April 2022 / Revised: 1 May 2022 / Accepted: 3 May 2022 / Published: 7 May 2022

Round 1

Reviewer 1 Report

Report on “NOVEL APPROACHES FOR DIFFERENTIABLE CONVEX
FUNCTIONS VIA THE PROPORTIONAL CAPUTO-HYBRID OPERATORS”

In this article, the authors studied some convexity properties related to classes of inequalities combined with fractional calculus. More precisely, they used the notion of proportional Caputo-hybrid operators for their main results by adapting differentiable convex functions using the indicated operators.

The work is interesting, reference list is complete and the theoretical results are applicable. I recommend the paper for publication after fixing the following points.

1/ Move Definitions 1-4 to Second section and reformulate the Introduction part again with clear motivation and contribution.

2/ Page 12, inequalities are not appearing properly, reformulate them.

3/ Abstract and conclusion section are almost similar, try to reconstruct.

4/ Include illustrative and/or numerical examples (with explicit expressions for the proportional Caputo-Hybrid Operators) to support the Abstract results.

Author Response

Dear editor,

We would like to thank to reviewer for the valuable suggestions.

  • We have arranged the introduction part and moved the definitions to the second part assuggested.
  • We have arranged the formulas for fixing to the page in page 12. 
  • We have re-written conclusion part as suggested. 
  • We did not add numerical examples. If we will give numerical examples, then we must choose K_0, K_1 functions and other parameters in special cases. In this case, clear truth statements will be achieved so that the variability underlying the fractional calculus will be overlooked.

Best regards

Reviewer 2 Report

Please see the attached file.

Comments for author File: Comments.pdf

Author Response

Dear editor,

We would like to thank to the reviewer for valuable suggestions. We have corrected the paper according to the report. 

  • We have corrected the formulas in page 6 and 12.
  • We have added a remark to demonstrate the special cases of K_0 and K_1. 
  • The acknowledgement section has been moved after to conclusion. 
  • The conclusion part has been arranged and we have added the reasons of the usage of this operator. 
  • The case for x=a+b/2 was already considered in the paper. 
  • We have added some recent papers to extend the introduction part.

Best regards

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