Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Please see the attached file.

Comments for author File: Comments.pdf

Author Response

Subject: Request for Reconsideration of Manuscript mathematics-4512691

Dear Editor,

We respectfully request reconsideration of the rejection decision for our manuscript, “Geometric Properties of the Normalized -Mittag--Leffler Function” (Manuscript ID: mathematics-4512691).

After carefully examining the reviewers' comments and comparing them with our original submitted manuscript, we have serious concerns regarding the basis of the mathematical assessment. Several criticisms are demonstrably inconsistent with the actual mathematical expressions and calculations contained in the original submission. In particular, some points identified as mathematical errors were already correctly presented in the submitted version, while another criticism resulted from a direct misinterpretation of a differentiation step.

We therefore respectfully request that the Editor independently verify these mathematical objections before relying on them as grounds for rejection. We are concerned that the manuscript may have been reviewed without sufficiently careful verification of the displayed calculations, rather than through a complete mathematical examination.

We fully respect the peer-review process, but given these serious discrepancies, we kindly request reconsideration of the decision and, if possible, an independent re-evaluation by another qualified expert in geometric function theory.

Thank you very much for your time and consideration.

Sincerely,
The Authors
Manuscript ID: mathematics-4512691

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The review report is attached.

Comments for author File: Comments.pdf

Author Response

Subject: Request for Reconsideration of Manuscript mathematics-4512691

Dear Editor,

We respectfully request reconsideration of the rejection decision for our manuscript, “Geometric Properties of the Normalized -Mittag--Leffler Function” (Manuscript ID: mathematics-4512691).

After carefully examining the reviewers' comments and comparing them with our original submitted manuscript, we have serious concerns regarding the basis of the mathematical assessment. Several criticisms are demonstrably inconsistent with the actual mathematical expressions and calculations contained in the original submission. In particular, some points identified as mathematical errors were already correctly presented in the submitted version, while another criticism resulted from a direct misinterpretation of a differentiation step.

We therefore respectfully request that the Editor independently verify these mathematical objections before relying on them as grounds for rejection. We are concerned that the manuscript may have been reviewed without sufficiently careful verification of the displayed calculations, rather than through a complete mathematical examination.

We fully respect the peer-review process, but given these serious discrepancies, we kindly request reconsideration of the decision and, if possible, an independent re-evaluation by another qualified expert in geometric function theory.

Thank you very much for your time and consideration.

Sincerely,
The Authors
Manuscript ID: mathematics-4512691

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Report attached.

Comments for author File: Comments.pdf

Author Response

Response to the Reviewer
Manuscript ID: mathematics-4512691
“On Geometric Properties of k-Mittag–Leffler Function”
Shahid Khan and Janusz Sok´o l
Dear Reviewer,
We thank you for your careful reading and helpful comments. All points have been addressed, and all
changes are shown in red in the revised manuscript.
Point 1. As late as on p. 5, Remark 1, it is stated “Since k > 0 throughout this paper” This should
be stated in the very beginning and in Abstract so that the reader doesn’t need to worry about the nature
of k.
Response: Done. k > 0 is now stated in the Abstract and in Section 1, just before the k-Pochhammer
symbol and k-Gamma function are introduced.
Point 2. Definition (13) of the k-Pochhammer symbol is not proper, since the variable may take
non-positive values.
Response: Agreed. To make this concept easier for the reader to follow, we have rewritten the
definition more clearly in the revised version. Definition (13) now has two parts ( Also see highlighted
part in revised version):
• Product form (always valid). For every υ ∈ C,
(υ)0,k = 1, (υ)r,k = υ(υ + k) · · ·
(︁υ + (r − 1)k)︁, r ∈ N.
This form is valid for every υ ∈ C, including non-positive values, because no Gamma function
appears in it.
• Gamma-quotient form (used only where it is valid). The formula (υ)r,k = Γk(υ +
rk)/Γk(υ) is used only when υ, υ + rk /∈ {0,−k,−2k, . . . }, i.e. away from the poles of Γk.
In our proofs the quotient form is needed only for q > 0 and real s ≥ 1, where every argument
of Γk is positive, so no difficulty arises. We also corrected the misprint in (24), which now reads
(υ)r,k = kr(υ/k)r, and added a one-line proof of it.
Point 3. In (26), the first term involves 1/Γk(ν), and ℜν > 0 may be necessary.
Response: Agreed. The normalization was already derived in the manuscript; however, we have
added more steps so that this point is clear to readers and reviewers. Please see the highlighted part
of revised version.
Point 4. The k-Gamma function seems to be defined only for ℜv > 0.
Response: Remark 1.1 now explains that (15) extends Γk meromorphically to C, with poles only
at 0,−k,−2k, . . . and no zeros. The recurrence is stated on this larger domain. In all proofs only
positive arguments occur.
1

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have provided a detailed response, and the concerns raised have been adequately addressed.

Reviewer 3 Report

Comments and Suggestions for Authors

Revision acceptable.