New General Variants of Chebyshev Type Inequalities via Generalized Fractional Integral Operators

Round 1

Reviewer 1 Report

The relevance of the original results is not clear. The results presented in the paper have very complicated forms,hence examples related to how they can be used must be provided. Please add examples of applications for the results. Most of the results are generalizations of previously known results or of themselves (Theorem 7 generalizes Theorem 6, Theorem 10 generalizes Theorem 9). Please explain in the paper how the results improve the already known results. There must exist some motivation you have had for the research. Also provide a conclusion containing some hints on how the results obtained in your paper are important for the theory of inequalities and how they could be used by researchers in the field for obtaining other results.

Author Response

ANSWERS TO REVIEWER COMMENTS

Firstly, we would like to thanks to reviewers and editors for their valuable comments on our paper. We have revised our paper according to their remarks.

The relevance of the original results is not clear. The results presented in the paper have very complicated forms, hence examples related to how they can be used must be provided. Please add examples of applications for the results. Most of the results are generalizations of previously known results or of themselves (Theorem 7 generalizes Theorem 6, Theorem 10 generalizes Theorem 9).
Please explain in the paper how the results improve the already known results. There must exist some motivation you have had for the research. Also provide a conclusion containing some hints on how the results obtained in your paper are important for the theory of inequalities and how they could be used by researchers in the field for obtaining other results.

• 
  We have explained the motivation of this paper to clarify the effectiveness of main results.
  Also, we have added a conclusion part as suggested to give the highlights.
• We have mentioned about application direction of our results in conclusion part.
• The results are clear generalizations of classical Chebyshev inequality.

Author Response File: Author Response.pdf

Reviewer 2 Report

1. Typos and EN

The MS contains lots of typos or bad EN, for example:

p.2, line 3: theirS -> their; in Def. 2: LetS -> Let us; “… coefficientS…. IS …” –>“ … ARE forming a bounded sequence..”;

p.3, in Rem. 3: “.. RANIA et al. [19]”-> “RAINA et al..”;

p.5, in mid: “both side..” -> “both sideS”;

Also, bad technicalities: -wrong punctuations – at many place there are full stops  before a sentence finishes (say before a displayed eq., like (1.1), etc); - missing empty intervals  (after .) etc.; in places math. symbols in text are not done in $.$ (say, p.5, 1^st line and next lines of Proof, p. 7, 1s line of Proof; etc); - many-many displayed eqs. go beyond the page width;

2. List of Refs:

There are observed some extra Ref items NOT related to the topic of inequalities in FC, and also not among the basic handbooks on FC theory. Say [7],[8],[9] need to be removed as not related and not giving good prestige for FC… Also, be careful on citations to works on Comformable “fractional” calculus operators, as such operators are definitely NOT of fractional order (as shown by many recent critical papers!), say  as [15], [28],… and some others on “new” (generalized) operators of FC that have been shown not correct…  In Ths. 1,2,3,4  (p.4) a result from [15] is cited for Coformable  “fractional” integrals, But the used symbols for I (with sub- and sup-indices \eta,\xi,\lambda) are NOT defined. Are these results necessary, if so – give explicit definition. Same for Th. 5.

3. Mathematical Contents:

A lot of long written evaluations and inequalities by using standard techniques. Such knd of functional inequalities are spread recently on many publications with changing the operators only, and as FC is now fashion, usually some “new” Fc operators are involved. The reviewer is not sure about their practical applications (although the Refs list tries to confirm this, by citing others – also not enough visible needs for).

Few words on the “generalized” k-fractional integrals (1.3) on which the study is based. In all such k-analogues, the notion of Gamma-k function is used, and authors claim to invent by its use some new special functions (SF), then – new operators of FC, etc. One has to note that the Gamma-k function is expressible well by mean of the classical Gamma functions (last expression in Def.1 !). Then, “new” k-SF  are representable Ok by means of the classical ones. Look at the SF in Def. 2. The main, “new” point is the sequence \sigma (m) but in next Remarks and examples it is not clear what is this sequence in some particular cases (except that \sigma(0)=1 in Remarks 5,6). What can be \sigma(m), m=1,2,… - example of such bounded sequence? At least, for \sigma = const or \sigma(m) \leq A (bounded), the function in Def. 2, for k=1 is, or bounded by the Mittag-Leffler function E_{\rho,\lambda), to be mentioned.

Conclusion: Better motivation and clarification of the results, and essential improvement of technicalities is necessary

Author Response

Firstly, we would like to thanks to the reviewer for the valuable comments on our paper. We have revised our paper according to their remarks.

1. Typos and EN

The MS contains lots of typos or bad EN, for example:

p.2, line 3: theirS -> their; in Def. 2: LetS -> Let us; “… coefficientS…. IS …” –>“ … ARE forming a

bounded sequence..”;

-We have corrected all typos.

p.3, in Rem. 3: “.. RANIA et al. [19]”-> “RAINA et al..”;

-Corrected.

p.5, in mid: “both side..” -> “both sideS”;

-Corrected

Also, bad technicalities: -wrong punctuations – at many place there are full stops before a sentence finishes (say before a displayed eq., like (1.1), etc); - missing empty intervals (after .) etc.; in places math. symbols in text are not done in $.$ (say, p.5, 1st line and next lines of Proof, p. 7, 1s line of Proof; etc); - many-many displayed eqs. go beyond the page width;

- We have checked whole of the paper in terms of punctuation marks, missing intervals and fullstops. We have corrected them.

2. List of Refs:

There are observed some extra Ref items NOT related to the topic of inequalities in FC, and also not among the basic handbooks on FC theory. Say [7],[8],[9] need to be removed as not related and not giving good prestige for FC… Also, be careful on citations to works on Comformable “fractional” calculus operators, as such operators are definitely NOT of fractional order (as shown by many recent critical papers!), say as [15], [28],… and some others on “new” (generalized) operators of FC that have been shown not correct… In Ths. 1,2,3,4 (p.4) a result from [15] is cited for Conformable “fractional” integrals. But the used symbols for I (with sub- and sup-indices \eta,\xi,\lambda) are NOT defined. Are these results necessary, if so – give explicit definition. Same for Th. 5.

- We have removed the aforementioned inappropriate references from the reference list.

- We have removed some unnecessary earlier results as suggested. They have proved by conformable and not directly related to our main findings.

3. Mathematical Contents:

A lot of long written evaluations and inequalities by using standard techniques. Such kind of functional inequalities are spread recently on many publications with changing the operators only, and as FC is now fashion, usually some “new” Fc operators are involved. The reviewer is not sure about their practical applications (although the Refs list tries to confirm this, by citing others – also not enough visible needs for).

• We have established some new integral inequalities of Chebyshev type via fractional integral operators. We know that it is hard to give applications to the results that have been established by fractional derivative and integrals. But, as we mentioned in the conclusion part, several applications can be given for special cases.

Few words on the “generalized” k-fractional integrals (1.3) on which the study is based. In all such k-analogues, the notion of Gamma-k function is used, and authors claim to invent by its use some new special functions (SF), then – new operators of FC, etc. One has to note that the Gamma-k function is expressible well by mean of the classical Gamma functions (last expression in Def.1 !). Then, “new” k-SF  are representable Ok by means of the classical ones. Look at the SF in Def. 2. The main, “new” point is the sequence \sigma (m) but in next Remarks and examples it is not clear what is this sequence in some particular cases (except that \sigma(0)=1 in Remarks 5,6). What can be \sigma(m), m=1,2,… - example of such bounded sequence? At least, for \sigma = const or \sigma(m) \leq A (bounded), the function in Def. 2, for k=1 is, or bounded by the Mittag-Leffler function E_{\rho,\lambda), to be mentioned.

• In definition 1-2, we have recalled well-known special functions that we have used them in the concluding remarks. We do not consider different choices of \sigma(m), because we only consider the cases that the operator overlap with another one.

Conclusion: Better motivation and clarification of the results, and essential improvement of technicalities is necessary.

• We have explained the motivation of this paper before the main results section and in conclusion section.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

 The requests have been complied as good as possible.

Reviewer 2 Report

Accept under the revision made, OK.