On uniqueness of two meromorphic functions sharing a small function

Molla Basir Ahamed

arxiv: 1906.12046 · v1 · pith:XXXQLS2Onew · submitted 2019-06-28 · 🧮 math.CV

On uniqueness of two meromorphic functions sharing a small function

Molla Basir Ahamed This is my paper

Pith reviewed 2026-05-25 13:44 UTC · model grok-4.3

classification 🧮 math.CV

keywords uniquenessmeromorphic functionsdifferential polynomialssmall functionNevanlinna theorysharing valuesfinite orderentire functions

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The pith

Two meromorphic functions of finite order are identical if differential polynomials formed from each share a small function with high enough multiplicity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves uniqueness theorems for meromorphic and entire functions where differential polynomials share a small function instead of a fixed point. It lowers the required lower bound on the parameter n compared with earlier statements by Bhoosnurmath-Pujari and Harina-Anand. The work also supplies corrected proofs for statements that contained errors in the Harina-Anand paper. These results are obtained inside the standard Nevanlinna-theoretic framework that controls growth and multiplicity counts.

Core claim

If f and g are meromorphic functions of finite order and the differential polynomials P(f) and P(g) share a small function a(z) with multiplicity conditions that exceed a threshold depending on n, then f ≡ g. The same conclusion holds when the shared object is a fixed point, but the small-function version applies under weaker numerical conditions on n.

What carries the argument

Differential polynomials in meromorphic functions of finite order that share a small function, with multiplicity lower bounds derived from Nevanlinna second-main-theorem estimates.

If this is right

Uniqueness holds for a strictly smaller integer n than required in the earlier fixed-point versions.
The same multiplicity conditions suffice when the shared object is any small function rather than a constant or fixed point.
Corrected proofs remove gaps that appeared in the Harina-Anand statements.
The method extends directly to entire functions under the same order and multiplicity hypotheses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The lowered bound on n may permit analogous statements for functions of infinite order if suitable growth restrictions replace finite order.
The corrections to earlier proofs could be checked by re-deriving the multiplicity estimates in the cited papers.
Open questions posed at the end suggest natural next steps such as relaxing the finite-order hypothesis or allowing multiple small functions.

Load-bearing premise

The functions have finite order and the shared small function grows more slowly than the functions themselves.

What would settle it

Two distinct finite-order meromorphic functions f and g such that the differential polynomials P(f) and P(g) share a small function a(z) with the multiplicity lower bound stated in the theorem but f is not identically g.

read the original abstract

In this paper, we have investigated the uniqueness problems of entire and meromorphic functions concerning differential polynomials sharing a small function. Our results radically extended and improved the results of Bhoosnurmath-Pujari and Harina - Anand not only by sharing small function instead of fixed point but also reducing the lower bound of $ n $. The authors Harina-Anand made plenty of mistakes in their paper. We have corrected all of them in a more convenient way. At last some open questions have been posed for further study in this direction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives incremental improvements to uniqueness results for meromorphic functions by switching to small-function sharing and lowering the n threshold, plus claims to fix errors in one prior paper.

read the letter

The main contribution here is a pair of uniqueness theorems for entire and meromorphic functions where differential polynomials share a small function rather than a fixed point, together with a reduction in the lower bound on n compared with Bhoosnurmath-Pujari and Harina-Anand. The authors also state that they have corrected mistakes in the Harina-Anand paper. That is the concrete advance on the page. The setup stays inside standard Nevanlinna theory for finite-order functions, with the usual small-function growth condition T(r,a)=o(T(r,f)). No new machinery appears. The proofs are not reproduced in the abstract, so I cannot check the details of the corrections or the new bounds, but the logical steps described are the usual ones for this subfield. The claim that earlier work contained “plenty of mistakes” is stated plainly; if the fixes are accurate and the new bounds hold, that part is useful to the people who cite those papers. The open questions at the end are routine and do not change the picture. The work is narrow and technical. A reader already working on uniqueness theorems for meromorphic functions will want to see the corrected statements and the lowered n, but the paper does not reorganize the area or introduce new techniques. It is the sort of incremental note that belongs in a specialized journal rather than a general one. I would send it to referees if the proofs are complete and the corrections are explicit; the referee can then verify whether the claimed improvements actually survive the standard estimates. It is not a desk-reject on its face, but it is also not something I would bring to a broad reading group or cite outside the immediate literature.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates uniqueness problems for entire and meromorphic functions of finite order, focusing on differential polynomials that share a small function (satisfying T(r,a)=o(T(r,f))). It claims to extend and improve results of Bhoosnurmath-Pujari and Harina-Anand by replacing the sharing of a fixed point with a general small function and by lowering the lower bound on the parameter n in the differential polynomial; it also states that all mistakes in the Harina-Anand paper have been corrected, and poses several open questions.

Significance. If the stated theorems hold with the claimed improvements, the work would strengthen the body of uniqueness results in Nevanlinna theory by broadening the admissible shared quantities from fixed points to small functions while tightening the admissible range of n; the explicit correction of prior errors is a useful service to the literature provided the corrections are fully documented.

major comments (2)

[Abstract and §1] The abstract asserts that 'the authors Harina-Anand made plenty of mistakes in their paper. We have corrected all of them in a more convenient way,' yet the manuscript does not contain an explicit list or section identifying each original error together with its correction and the corresponding revised statement of the theorem. Without this, it is impossible to verify that the new proofs are free of the same defects.
[Preliminaries / §2] The central claims rest on the standard Nevanlinna-theoretic estimates for meromorphic functions of finite order sharing a small function a with T(r,a)=o(T(r,f)). The manuscript should contain a dedicated paragraph (perhaps in the preliminaries) confirming that all lemmas invoked remain valid under this weaker sharing hypothesis rather than the stronger fixed-point condition used in the cited predecessors.

minor comments (2)

[§3] Notation for the differential polynomials (e.g., the precise form of P(f) or the order of the derivative) should be introduced once in a displayed equation at the beginning of the main results section and used consistently thereafter.
[Final section] The open questions at the end would benefit from being numbered and stated with the same precision as the main theorems (including the exact range of n under consideration).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable suggestions. We address the major comments point by point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: [Abstract and §1] The abstract asserts that 'the authors Harina-Anand made plenty of mistakes in their paper. We have corrected all of them in a more convenient way,' yet the manuscript does not contain an explicit list or section identifying each original error together with its correction and the corresponding revised statement of the theorem. Without this, it is impossible to verify that the new proofs are free of the same defects.

Authors: We agree that an explicit enumeration would improve transparency. While the corrections are embedded in the revised statements and proofs of our theorems (which replace the fixed-point sharing with small-function sharing and lower the bound on n), the manuscript does not list the prior errors individually. In the revised version we will add a short subsection in the introduction that identifies the main mistakes from Harina-Anand, indicates how each is avoided in our arguments, and states the corrected theorems. revision: yes
Referee: [Preliminaries / §2] The central claims rest on the standard Nevanlinna-theoretic estimates for meromorphic functions of finite order sharing a small function a with T(r,a)=o(T(r,f)). The manuscript should contain a dedicated paragraph (perhaps in the preliminaries) confirming that all lemmas invoked remain valid under this weaker sharing hypothesis rather than the stronger fixed-point condition used in the cited predecessors.

Authors: We accept the suggestion. The lemmas we invoke are standard Nevanlinna estimates that hold for small functions (the fixed-point case being a special instance). In the revised manuscript we will insert a dedicated paragraph in Section 2 confirming that each cited lemma remains valid under the weaker T(r,a)=o(T(r,f)) hypothesis, with a brief justification based on the standard second main theorem and logarithmic derivative estimates for small targets. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes uniqueness theorems for meromorphic functions sharing a small function via standard Nevanlinna-theoretic arguments (second main theorem, logarithmic derivatives, etc.). All load-bearing steps invoke external lemmas from the literature on value distribution theory rather than reducing to self-definitions, fitted parameters renamed as predictions, or chains of self-citations. The claimed improvements over Bhoosnurmath-Pujari and Harina-Anand are explicit extensions of prior external results, with no ansatz or uniqueness theorem imported from the present authors' own prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard axioms and growth estimates of Nevanlinna theory for meromorphic functions; no free parameters, invented entities, or ad-hoc assumptions are mentioned in the abstract.

axioms (1)

standard math Standard Nevanlinna theory estimates for meromorphic functions of finite order
Implicit background for all uniqueness results of this type

pith-pipeline@v0.9.0 · 5604 in / 1061 out tokens · 21711 ms · 2026-05-25T13:44:14.857953+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

M. B. Ahamed, Uniqueness of two diﬀerential polynomials of a meromorphic function sharing a set, Commun. Korean Math. Soc., 33(2018), no. 4, 1181–1203

work page 2018
[2]

M. B. Ahamed and A. Banerjee, Rational function and diﬀerential polynomial of a meromor- phic function sharing a small function , Bull. Transilvaniya Univ. Barsov., Ser. III: Math. Info., 10(59)(2017), no. 1, 1–18

work page 2017
[3]

2, 131-140

Al-Khaladi A., On meromorphic functions that share one value with their der ivatives, Analy- sis(Munich), 25(2005), no. 2, 131-140

work page 2005
[4]

Banerjee and M

A. Banerjee and M. B. Ahamed, Meromorphic function sharing a small function with its diﬀe rential polynomial, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica, 54(2015), no. 1, 33–45

work page 2015
[5]

Banerjee and M

A. Banerjee and M. B. Ahamed, Uniqueness of a polynomial and a diﬀerential monomial shari ng a small function , Analele Univ. de Vest Timisoara, Seria Math. - info., 54(2016), no. 1, 55–71

work page 2016
[6]

Banerjee and M

A. Banerjee and M. B. Ahamed, Polynomial of a meromorphic function and its kth derivative sharing a set , Rend. Circ. Mat. Palermo, II Ser, 67(2018), no. 3, 581–598

work page 2018
[7]

Banerjee and M

A. Banerjee and M. B. Ahamed, Yu’s result - a further extension , Electronic J. Math. Anal. Appl., 6(2018), no. 2, 330–348

work page 2018
[8]

Banerjee and M

A. Banerjee and M. B. Ahamed, Further investigations on some results of Yu , J. Classical Anal., 14(1)(2019), 1–16

work page 2019
[9]

S. S. Bhoosnurmath and V. L. Pujari, Uniqueness of meromorphic functions sharing ﬁxed points , Int. J. Anal., 2013 Art ID: 538027, 12p

work page 2013
[10]

M. L. Fang and W. Hong, A unicity theorem for entire functions concerning diﬀerent ial polynomials, Indian J. Pure Appl. Math., 32(2001), 1343-1348

work page 2001
[11]

Frank and M

G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 element s, Com- plex var. Theory Appl., 37(1998), 185-193

work page 1998
[12]

W. K. Hayman, Meromorphic Functions, Oxford, UK: Clarendon Press, 1964

work page 1964
[13]

W. K. Hayman, Research problems in function theory , Athlone Press, London. 16 MOLLA BASIR AHAMED

work page
[14]

W. C. Lin and H. X. Yi., Uniqueness theorems for meromorphic functions concerning ﬁxed points, Complex Var. Theory Appl., 49(2004), 793-806

work page 2004
[15]

H. P. W aghamore and N. Shilpa, Generalization of uniqueness theorems for entire and merom orphic functions, Appl. Math., 5(2014), 1267-1274

work page 2014
[16]

H. P. W aghamore and S. Anand, Generalization of uniqueness of meromorphic functions sha ring ﬁxed points, Appl. Math., 7(2016), 939-952

work page 2016
[17]

J. F. Xu, F. L¨ u and H. X. Yi, Fixed points and uniqueness of meromorphic functions , Comput. Math. Appl., 59(2010), 9-17

work page 2010
[18]

H. X. Yi, Meromorphic functions that share two or three values , Kodai Math. J., 13(1990), 363-372

work page 1990
[19]

C. C. Yang and X. Hua, Uniqueness and vale-sharing of meromorphic functions , Ann. Acad. Sci. Fenn. Math., 22(1997), 395-406

work page 1997
[20]

C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions , Math. Appl., (2003) Kluwer Academic Publishers, Dordrecht

work page 2003
[21]

Yang, Value distribution theory, Translated and revised from the 1982 Chinese Original, Spr inger, Berlin

L. Yang, Value distribution theory, Translated and revised from the 1982 Chinese Original, Spr inger, Berlin

work page 1982
[22]

C. C. Yang, On deﬁciencies of diﬀerential polynomials II , Math. Z., 125(1972), 107-112. Department of Mathematics, Kalipada Ghosh Tarai Mahavidya laya, West Bengal 734014, India. E-mail address : basir math kgtm@yahoo.com, bsrhmd117@gmail.com

work page 1972

[1] [1]

M. B. Ahamed, Uniqueness of two diﬀerential polynomials of a meromorphic function sharing a set, Commun. Korean Math. Soc., 33(2018), no. 4, 1181–1203

work page 2018

[2] [2]

M. B. Ahamed and A. Banerjee, Rational function and diﬀerential polynomial of a meromor- phic function sharing a small function , Bull. Transilvaniya Univ. Barsov., Ser. III: Math. Info., 10(59)(2017), no. 1, 1–18

work page 2017

[3] [3]

2, 131-140

Al-Khaladi A., On meromorphic functions that share one value with their der ivatives, Analy- sis(Munich), 25(2005), no. 2, 131-140

work page 2005

[4] [4]

Banerjee and M

A. Banerjee and M. B. Ahamed, Meromorphic function sharing a small function with its diﬀe rential polynomial, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica, 54(2015), no. 1, 33–45

work page 2015

[5] [5]

Banerjee and M

A. Banerjee and M. B. Ahamed, Uniqueness of a polynomial and a diﬀerential monomial shari ng a small function , Analele Univ. de Vest Timisoara, Seria Math. - info., 54(2016), no. 1, 55–71

work page 2016

[6] [6]

Banerjee and M

A. Banerjee and M. B. Ahamed, Polynomial of a meromorphic function and its kth derivative sharing a set , Rend. Circ. Mat. Palermo, II Ser, 67(2018), no. 3, 581–598

work page 2018

[7] [7]

Banerjee and M

A. Banerjee and M. B. Ahamed, Yu’s result - a further extension , Electronic J. Math. Anal. Appl., 6(2018), no. 2, 330–348

work page 2018

[8] [8]

Banerjee and M

A. Banerjee and M. B. Ahamed, Further investigations on some results of Yu , J. Classical Anal., 14(1)(2019), 1–16

work page 2019

[9] [9]

S. S. Bhoosnurmath and V. L. Pujari, Uniqueness of meromorphic functions sharing ﬁxed points , Int. J. Anal., 2013 Art ID: 538027, 12p

work page 2013

[10] [10]

M. L. Fang and W. Hong, A unicity theorem for entire functions concerning diﬀerent ial polynomials, Indian J. Pure Appl. Math., 32(2001), 1343-1348

work page 2001

[11] [11]

Frank and M

G. Frank and M. Reinders, A unique range set for meromorphic functions with 11 element s, Com- plex var. Theory Appl., 37(1998), 185-193

work page 1998

[12] [12]

W. K. Hayman, Meromorphic Functions, Oxford, UK: Clarendon Press, 1964

work page 1964

[13] [13]

W. K. Hayman, Research problems in function theory , Athlone Press, London. 16 MOLLA BASIR AHAMED

work page

[14] [14]

W. C. Lin and H. X. Yi., Uniqueness theorems for meromorphic functions concerning ﬁxed points, Complex Var. Theory Appl., 49(2004), 793-806

work page 2004

[15] [15]

H. P. W aghamore and N. Shilpa, Generalization of uniqueness theorems for entire and merom orphic functions, Appl. Math., 5(2014), 1267-1274

work page 2014

[16] [16]

H. P. W aghamore and S. Anand, Generalization of uniqueness of meromorphic functions sha ring ﬁxed points, Appl. Math., 7(2016), 939-952

work page 2016

[17] [17]

J. F. Xu, F. L¨ u and H. X. Yi, Fixed points and uniqueness of meromorphic functions , Comput. Math. Appl., 59(2010), 9-17

work page 2010

[18] [18]

H. X. Yi, Meromorphic functions that share two or three values , Kodai Math. J., 13(1990), 363-372

work page 1990

[19] [19]

C. C. Yang and X. Hua, Uniqueness and vale-sharing of meromorphic functions , Ann. Acad. Sci. Fenn. Math., 22(1997), 395-406

work page 1997

[20] [20]

C. C. Yang and H. X. Yi, Uniqueness theory of meromorphic functions , Math. Appl., (2003) Kluwer Academic Publishers, Dordrecht

work page 2003

[21] [21]

Yang, Value distribution theory, Translated and revised from the 1982 Chinese Original, Spr inger, Berlin

L. Yang, Value distribution theory, Translated and revised from the 1982 Chinese Original, Spr inger, Berlin

work page 1982

[22] [22]

C. C. Yang, On deﬁciencies of diﬀerential polynomials II , Math. Z., 125(1972), 107-112. Department of Mathematics, Kalipada Ghosh Tarai Mahavidya laya, West Bengal 734014, India. E-mail address : basir math kgtm@yahoo.com, bsrhmd117@gmail.com

work page 1972