On Special Fuzzy Differential Subordinations Obtained for Riemann–Liouville Fractional Integral of Ruscheweyh and Sălăgean Operators

Round 1

Reviewer 1 Report

The study of geometric functions theory in the context of  Fuzzy Theory is quite interesting and relatively new concept which merit itself for possible publications in Axiom. However, since there are only few articles in the literature, author needs to provide more careful, clear description with example(s) of each obtained results. Beside, a connection between the said development of fuzzy subordination theory with classical subordination (in general Geometric/Univalent functions theory )  is required. After careful reading, it is clear that all the stated result with proof are correct. However, author should address following before final consideration: 

1. Example with details justifications for each of the obtained results. One Example is provided but the justification/verification is not clear. Please provide more details. A graphical presentation will be more useful. 

2. A connection between the new development "Fuzzy Subordination Theory" with "Classical Subordination Theory" (Atleast based on the examples stated in comment (1)) . 

Author Response

The response is in the attached file.

Author Response File: Author Response.docx

Reviewer 2 Report

Reviewer Comments

Title: On special fuzzy differential subordinations obtained for Riemann–Liouville fractional integral of Ruscheweyh and Sălăgean operators

Authors: Alina Alb Lupaş

Journal Name:  Axioms (ISSN 2075-1680)

ID: axioms-1846964

The paper needs some revisions from the point of authors and readers to improve the quality of the paper. After these “MINOR REVISIONS”, I suggest that this paper can be resubmitted to review again in " Axioms (ISSN 2075-1680) ".

My comments are as follows:

1. Ensure the end of each line of the equations has punctuation, either a comma or a full stop if it is the end of the equation - these are missing in several of the equations, see for example; Eqs in Definition 4.

2. Several grammatical and typo errors are stated. They need to be removed. For example: "Taking an holomorphic function...", "Considering an holomorphic function..."

Line 160-161: It can be introduce other subclasses of analytic functions regarding this operator and it can be investigate some properties..."

3. It seems that the literature investigations are weak. They need to improve the paper by comparing the results given in the related literature.

4. The paper contains some basic mathematical analysis without any reason or interpretation.

5. The conclusion section should be given to include the paper's main findings and future direction. 

Author Response

Response to Reviewer 2 Comments

Thank you so much for carefully reading the paper and pointing out all the parts that were not so well written. The changes suggested by you have considerably improved the form of the paper and its correctness! Changes are highlighted in the text.

The paper needs some revisions from the point of authors and readers to improve the quality of the paper. After these “MINOR REVISIONS”, I suggest that this paper can be resubmitted to review again in " Axioms (ISSN 2075-1680) ".

My comments are as follows:

1. Ensure the end of each line of the equations has punctuation, either a comma or a full stop if it is the end of the equation - these are missing in several of the equations, see for example; Eqs in Definition 4.

 Response 1: Done

2. Several grammatical and typo errors are stated. They need to be removed. For example: "Taking an holomorphicfunction...", "Considering an holomorphic function..."

Line 160-161: It can be introduce other subclasses of analytic functions regarding this operator and it can be investigate some properties..."

 Response 2: Done

3. It seems that the literature investigations are weak. They need to improve the paper by comparing the results given in the related literature.

 Response 3: The results presented in this paper can’t be compared to other results since this is the only paper which deals with fuzzy differential subordinations involving the Riemann-Liouville fractional integral applied to the convex combination of well-known Ruscheweyh and Sălăgean differential operators.

However, we have stated in lines 75-80 of the paper:

“New operators introduced using fractional integral and applied in fuzzy differential subordination theory were studied in [17] where Atangana–Baleanu fractional integral is applied, in [18] where Riemann-Liouville fractional integral is applied for Gaussian hypergeometric function and in [19] where Riemann-Liouville fractional integral is combined with confluent hypergeometric function.” Hence, a comparison is made in the only possible way citing similar operators used by other authors.

Also, in lines 85-87 of the paper we show that “In the next section, a new fuzzy class will be defined and studied in order to obtain fuzzy differential subordinations inspired by recently published researches concerned with the same topic seen in [20]-[22].” Three papers can be read by the interested researchers and see the difference between the results presented there and the ones contained here, a difference which appears due to the use of a different operator in this paper.

4. The paper contains some basic mathematical analysis without any reason or interpretation.

Response 4: All the definitions and notations given in Introduction are taken from cited published papers which means they are known to the researchers in the field and correct. They are all used in the study presented in this paper, hence they must be given here. The results presented in Main Results are new fuzzy differential subordinations for which the best fuzzy dominants are found which is the main focus in the research related to this topic. They can’t be explained, the reason for obtaining those results is the development of the theory itself!

5. The conclusion section should be given to include the paper's main findings and future direction. 

Response 5: We have rephrased the Conclusion part making it obvious where the main findings of the present paper are presented and where future directions are given:

“          In the study presented in this paper, a new subclass of analytic functions denoted by  is introduced in Definition 9 using the operator given in Definition 8. The new subclass  is investigated by applying the theory of fuzzy differential subordination, and several interesting properties are proved such as its convexity and some inclusion relations concerning the parameters used in the definition of the class. New fuzzy differential subordinations are obtained in the theorems involving the operator  for which fuzzy best dominants are found. In order to show how the results could be applied, an example is also given.

Future studies can be done regarding the operator introduced in Definition 8 and the subclass  introduced in Definition 9. Using the operator , other subclasses of analytic functions could be defined and the classes could be investigated using different aspects of the theory of fuzzy differential subordination. The new subclass  can be further investigated in order to obtain coefficient estimates, closure theorems, distortion theorems, neighbourhoods and the radii of starlikeness, convexity or close-to-convexity.

Another future direction of study is given by the possibility of using the dual theory of fuzzy differential superordination introduced in [23] for obtaining fuzzy differential superordinations involving the operator   and the class  which could be combined with the results presented here for sandwich-type theorems as seen in [19].”

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The article in current form address some of the concerns raised in earlier review.  There is some improvement in  the article, however few minor editing require. 

1. Example 1, 3, 4 are based on same function g(z). So, there is no requirement to show the derivation of h(z) in all examples. Directly, write " consider g(z) and h(z) as given in Example 1. Then........"  

2. Please write big equation, almost in all example,  in equation mode \[ \]. 

Author Response

Response to Reviewer 1 Comments

Thank you so much for carefully reading the paper and pointing out all the parts that were not so well written. The changes suggested by you have considerably improved the form of the paper and its correctness! Changes are highlighted in the text.

The article in current form address some of the concerns raised in earlier review.  There is some improvement in  the article, however few minor editing require. 

1. Example 1, 3, 4 are based on same function g(z). So, there is no requirement to show the derivation of h(z) in all examples. Directly, write " consider g(z) and h(z) as given in Example 1. Then........"  

Response 1: The correction has been done.

2. Please write big equation, almost in all example,  in equation mode \[ \]. 

Response 2: The correction has been done.