Vanishing, Unbounded and Angular Shifts on the Quotient of the Difference and the Derivative of a Meromorphic Function

Lasse Asikainen; Risto Korhonen; Yu Chen

REVIEW 4 major objections 4 minor 12 references

Vanishing, Unbounded and Angular Shifts on the Quotient of the Difference and the Derivative of a Meromorphic Function

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that for meromorphic functions, the proximity function of the quotient of a finite-difference operator (with vanishing, slowly growing, or angular shift) and the derivative is small—zero in the vanishing case, $S(r,f')$…

desk verdict Main theorems are false as stated: rational functions satisfy the hypotheses but the S(r,f') conclusion fails; the variable-shift idea may be salvageable with an added growth condition. read the letter →

arxiv 2505.21150 v1 pith:UA7BGR5I submitted 2025-05-27 math.CV

Lasse Asikainen , Yu Chen , Risto Korhonen This is my paper

classification math.CV MSC 30D35

keywords meromorphicfunctionsvariableshiftNevanlinnatheoryproximityfunctionhyper-orderfinitedifferenceangulardeficiencyrelations

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that for a meromorphic function, the quotient of a finite-difference operator and the derivative is asymptotically close to 1 in the sense of Nevanlinna proximity functions, provided the shift is small, slowly growing, or a small angular rotation. Specifically, it claims that as the period $\eta$ tends to zero, the proximity function $m_\eta(r, \frac{\Delta_\eta f - a\eta}{f' - a})$ tends to zero; when the period $\omega$ grows no faster than $r^\beta$ with hyper-order less than $\frac34$, the proximity function is $S(r,f')$ outside a small exceptional set; and for an angular shift, the analogous proximity function is $S(r,f)$. These estimates matter because they connect the value distribution of finite differences to that of the derivative, yielding deficiency relations and bounds on separated pair indices.

What carries the argument

The load-bearing object is the proximity function $m\left(r, \frac{\Delta_\omega f - a\omega}{f' - a}\right)$ together with the Poisson–Jensen representation of $\log|g'|$ on circles, which converts the integrated oscillation of the shift into weighted sums over zeros and poles. The estimates are carried by a quotient $R_{\epsilon,\omega}(r,g)$ of angular counting functions, and by Lemma 2.2, a variant of the growth lemma cited as [9] that controls $T(r+\omega) - T(r)$ for hyper-order less than 1 when $|\omega| < r^\beta$. The mean value theorem gives the initial integral identity; the angular-shift case uses a similar identity with the rotation $e^{i\omega(r)}$.

What would settle it

For $f(z)=z^2$ and $\omega(r)=r^\beta$ with $0<\beta<\frac12$, the quotient $\frac{\Delta_\omega f}{f'}$ equals $\omega + \frac{\omega^2}{2z}$, so $m\left(r, \frac{\Delta_\omega f}{f'}\right) = \beta \log r + O(1)$, while $T(r,f') \sim \log r$; because $\beta \log r$ is not $o(\log r)$, the claimed equality $m = S(r,f')$ fails even though $f$ has hyper-order 0 and satisfies the theorem's hypotheses.

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Extended reading notes

Core claim

On its own terms, the central discovery is that the quotient $\frac{\Delta_\omega f - a\omega}{f' - a}$ has Nevanlinna proximity function $S(r,f')$ for variable shifts, not just fixed shifts. The proof represents the quotient as an integral of $f'(z+u)/f'(z)$ by the mean value theorem, then uses the Poisson–Jensen formula to bound the integrated logarithm by sums over zeros and poles of $f' - a$. A key growth lemma bounds $T(r+\omega) - T(r)$ for hyper-order less than 1. The angular-shift version uses the same Poisson–Jensen oscillation machinery with a rotation $e^{i\omega(r)}$ replacing translation by $\omega(r)$.

Load-bearing premise

The proof treats the error term $O(\log r)$ as negligible compared with $T(r,f')$, which is justified only when $T(r,f')$ grows faster than $\log r$; without that assumption, the $S(r,f')$ conclusion can fail for rational functions.

Editorial extensions

If this is right

The vanishing-shift estimate implies the proximity function tends to zero both as $\eta \to 0$ and as $r \to \infty$ when $|\eta|$ is smaller than any prescribed $\alpha(r)$ vanishing at infinity.
For unbounded shifts with $0 < |\omega(r)| < r^\beta$ and hyper-order $\varsigma < \frac34$, the quotient's proximity function is small, giving deficiency inequalities such as $\delta(a,f') \le (1 + \limsup N(r,f)/T(r,f'))\, \delta(a\omega, \Delta_\omega f)$.
For entire functions, the separated-pair indices $\pi_\eta(a,f)$ and $\pi_\omega(a,f)$ sum to at most $1 - \delta(0,f')$, mirroring the classical second main theorem bound.
The angular shift estimate yields the same $S(r,f)$ proximity bound for rotations, extending the theory to non-translation shifts.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The theorems are stated for all non-constant meromorphic functions, but the proof's use of $O(\log r) = S(r,f')$ implicitly requires transcendental growth; a corrected statement would likely add $T(r,f')/\log r \to \infty$.
The angular-shift result may extend to shifts that are powers of $r$ or to several angular parameters, since the Poisson–Jensen oscillation bound only needs $|e^{i\omega(r)} - 1|$ small.
The vanishing-shift limit suggests a local normal-family interpretation: as $\eta \to 0$, the difference quotient converges to $f'/f' = 1$ away from singularities, which is consistent with the proximity limit.
The $R_{\epsilon,\omega}$ angular counting ratio suggests a quantitative refinement: functions whose poles or zeros are concentrated near a small angular sector will make the main terms larger, so the bound degrades exactly when angular concentration is high.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The paper is not ready for publication. The central estimates (Theorems 1.2 and 1.3) are false as stated. The proof's final step turns an explicit +O(log r) into S(r,f'), which only holds when T(r,f')/log r → ∞. Rational functions satisfy the stated hypotheses (hyper-order 0) but miss this. For f(z)=z^2 and ω(r)=r^β, m(r,(Δ_ω f)/(f')) is β log r + O(1) while T(r,f')~log r, so m/T does not tend to 0. Theorem 1.3 fails similarly for f(z)=z.

That said, the paper has some genuine ideas. The variable-shift framework and the angular counting function R_{ε,ω} are new relative to the fixed-shift result in [1], and the Poisson-Jensen estimates in Section 4 are worked out in detail. The vanishing-shift result (Theorem 1.1) is basically a mean-value-theorem observation, but it may be useful as a lemma.

The main problem is not a missing detail but a missing hypothesis. If the authors add the condition that f is transcendental (or more precisely T(r,f')/log r → ∞), the proof might go through for transcendental functions. The reliance on [1, Lemma 3.1] from an unpublished preprint is also a concern, but the bigger issue is the false statement.

There are also several smaller gaps and typos (e.g., the definition of R_{ε,ω} is a bit terse, and the handling of exceptional sets gets muddled in places). I would recommend rejecting the current version. The author(s) should correct the theorem statements and verify the estimates under a sound growth condition; a revised version that proves the transcendental case cleanly could be worth re-submitting.

I would not send this to a referee as is. The counterexample is immediate, and the referee would come back with a rejection.

Referee Report

4 major / 4 minor

Summary. The paper studies proximity functions of quotients of finite difference operators and derivatives of meromorphic functions. For a vanishing shift, Theorem 1.1 claims limits of m_η(r,(Δ_η f - aη)/(f'-a)) as η→0 and as r→∞ under a pointwise small shift. Theorem 1.2 claims that for a meromorphic function of hyper-order ς<3/4 and a shift ω(r) with 0<|ω(r)|<r^β for a suitable β, the proximity function m(r,(Δ_ω f - aω)/(f'-a)) equals S(r,f') outside an exceptional set of finite logarithmic measure. Theorem 1.3 makes an analogous claim for the angular shift f(e^{iω(r)}z)-f(z) over f'. Section 6 derives deficiency relations and results on η/ω-separated pair indices from these estimates.

Significance. If Theorem 1.2 were correct, it would be a substantial extension of fixed-step difference Nevanlinna theory to variable shifts, with explicit error terms and new applications to deficiencies and separated pair indices. The paper is ambitious and contains quantitative estimates. However, the main theorem is false as stated, and the applications in Section 6 therefore do not follow. The paper does not provide machine-checked proofs or code, and its central proof depends on an unpublished preprint by two of the same authors.

major comments (4)

[Section 4, Theorem 1.2] The theorem is false as stated. The proof ends at (4.16) with a term O(log r) and then asserts that this is S(r,f'), which requires log r = o(T(r,f')). The hypotheses admit rational functions, for which T(r,f')~log r. For f(z)=z^2, a=0, and ω(r)=r^β with 0<β<1/2 (allowed because ς=0 and β<min{1/2,1}), the quotient (Δ_ω f)/(f') equals ω + ω^2/(2z), so m(r,(Δ_ω f)/(f')) ~ β log r, while T(r,f')~log r. Hence m/T → β ≠ 0, contradicting the claimed S(r,f'). The theorem needs an additional hypothesis such as f transcendental, equivalently T(r,f')/log r → ∞.
[Section 4, equations (4.9)-(4.10)] The proof of Theorem 1.2 relies on [1, Lemma 3.1], an unpublished preprint by two of the same authors, for the key estimates of the Z1 and P1 terms. This lemma is not stated or proved in the manuscript, and the main result depends on it; the proof is therefore not self-contained. The authors should either include a proof of the lemma or cite a published, verifiable source.
[Section 3, Theorem 1.1] The proof of Theorem 1.1 is incomplete. The inequality (3.2) and the statement that the quotient is <1 when η→0 do not establish the claimed limits, because the quotient can be large near zeros of f'-a and the set of θ where this happens must be estimated. The proof should quantify the measure of the exceptional intervals; as written, the assertion 'we see that |...|<1' is not sufficient to justify the limits of the integrated proximity function.
[Section 5, Theorem 1.3] The proof of Theorem 1.3 appears to use a bound involving T(r+ε,f') in (5.4) rather than the T(r+ε,f) that appears in the theorem's hypothesis on ω(r). Also, the condition 'for any ε>0' is not exploited systematically; if interpreted literally it forces ω(r) to be at most r^{-1+o(1)}, which is much stronger than the single-ε bounds used in parts of the proof. The derivation of S(r,f) from these estimates should be reworked.

minor comments (4)

[Section 1, Theorem 1.2] The definition of R_{ε,ω}(r,g) contains the expression sin(arg z0/ω); when ω is complex this is not well-defined and should be replaced by an angular separation condition such as |arg z0 - arg ω|.
[Section 3] The notation m_η(r,·) is not defined; it seems to denote the proximity function with the shift η fixed, but this should be stated explicitly.
[Section 4, equation (4.2)] In applying the Poisson-Jensen formula to log|g'(z)|, the constant term log|g'(0)| appears to be missing from the displayed formula.
[Section 6] There are several typographical issues, including 'the same as be the same as defined' in Proposition 6.5 and inconsistent notation for counting functions such as n_{∠,ε,ω}(r,g) before its definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main estimates are derived by direct Poisson–Jensen and standard Nevanlinna lemmas, and the only salient self-citation is a technical lemma that is not shown to be a restatement of the target results.

full rationale

The derivation chain is not circular in the sense of reducing a claimed prediction to an input by construction. Theorem 1.1 follows immediately from the mean value theorem. The proofs of Theorems 1.2 and 1.3 are explicit Poisson–Jensen estimates, using the standard Lemma 2.1 from Halburd–Korhonen–Tohge, the paper's own Lemma 2.2 (proved in Section 2), and the growth estimate [4, Lemma 3.3.1]. The one self-citation is the use of [1, Lemma 3.1] for the angular-sector contributions in (4.9)–(4.10). That lemma is load-bearing for the terms involving n∠ε,ω, and [1] is an unpublished preprint co-authored by two of the present authors. However, the paper does not define its main conclusion in terms of that lemma, and nothing in the quoted passage indicates that [1, Lemma 3.1] is equivalent to, or assumed as, the theorem being proved. It is a technical potential-theoretic estimate, so this is a citation dependency rather than a circular reduction. The false-as-stated issue for rational functions, where T(r,f')∼log r and the final O(log r)=S(r,f') step fails, is a missing growth assumption and therefore a correctness problem, not a circularity problem. Accordingly, no step in the claimed derivation chain reduces to its own input, and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters fitted to data and no invented physical entities. It relies on standard Nevanlinna theory, two cited lemmas from prior work, and an implicit transcendence assumption that is missing from the theorem statements.

assumptions (6)

standard math Nevanlinna theory and Poisson-Jensen formula
Used throughout Sections 4 and 5 for proximity function estimates.
standard math Lemma 2.1 (Halburd, Korhonen, Tohge, Lemma 8.3)
Controls difference quotients of T(r,f) for hyper-order <1; invoked in the proof of Theorem 1.2.
domain assumption [1, Lemma 3.1]
A key estimate for sums over zeros and poles near the circle, taken from an unpublished preprint by overlapping authors; it is not re-proved here.
standard math [4, Lemma 3.3.1] (Cherry-Ye)
Used to absorb the α(r) factor in T(α(r+|ω|),g').
domain assumption Hyper-order ς<3/4 and 0<β<min{(1-ς)/2, 1-4ς/3}
Growth and shift restrictions in Theorem 1.2; ensure the error exponents are positive.
ad hoc to paper Transcendence of f
The proof's final step requires T(r,f')/log r→∞ so that O(log r)=o(T(r,f')); this is not stated in the theorems.

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Cite this review

Pith. "Pith review of Vanishing, Unbounded and Angular Shifts on the Quotient of the Difference and the Derivative of a Meromorphic Function." pith.science (2026). https://pith.science/paper/UA7BGR5I

@misc{pith2026250521150,
  author       = {Pith},
  title        = {Pith review of: Vanishing, Unbounded and Angular Shifts on the Quotient of the Difference and the Derivative of a Meromorphic Function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UA7BGR5I}},
  note         = {Machine review of arXiv:2505.21150}
}

abstract

We show that for a vanishing period difference operator of a meromorphic function $ f $, there exist the following estimates regarding proximity functions, \[ \lim_{\eta \to 0} m_\eta\left(r, \frac{\Delta_\eta f - a\eta}{f' - a} \right) = 0 \] and \[ \lim_{r \to \infty} m_\eta\left(r, \frac{\Delta_\eta f - a\eta}{f' - a} \right) = 0, \] where $ \Delta_\eta f = f(z + \eta) - f(z) $, and $ |\eta| $ is less than an arbitrarily small quantity $ \alpha(r) $ in the second limit. Then, under certain assumptions on the growth, restrictions on the period tending to infinity, and on the value distribution of a meromorphic function $ f(z) $, we have \[ m\left(r, \frac{\Delta_\omega f - a\omega}{f' - a} \right) = S(r, f'), \] as $ r \to \infty $, outside an exceptional set of finite logarithmic measure. Additionally, we provide an estimate for the angular shift under certain conditions on the shift and the growth. That is, the following Nevanlinna proximity function satisfies \[ m\left(r, \frac{f(e^{i\omega(r)}z) - f(z)}{f'} \right) = S(r, f), \] outside an exceptional set of finite logarithmic measure. Furthermore, the above estimates yield additional applications, including deficiency relations between $ \Delta_\eta f $ (or $ \Delta_\omega f $) and $ f' $, as well as connections between $ \eta/\omega $-separated pair indices and $ \delta(0, f') $.

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

A new proximity function estimate on the quotient of the difference and the derivative of a meromorphic function

L. Asikainen, J. M. Huusko, and R. Korhonen. A new proximi ty function estimate on the quotient of the diﬀerence and the derivative of a meromorphi c function. arXiv preprint arXiv:2306.06729, 2023

work page Pith review arXiv 2023
[2]

Bergweiler and J

W. Bergweiler and J. K. Langley. Zeros of diﬀerences of me romorphic functions. Math. Proc. Cambridge Philos. Soc. , 142(1):133–147, 2007

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[3]

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Y. M. Chiang and S. J. Feng. On the growth of logarithmic di ﬀerences, diﬀerence quotients and logarithmic derivatives of meromorphic functions. Trans. Amer. Math. Soc., 361(7):3767– 3791, 2009

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Y. M. Chiang and X. D. Luo. Diﬀerence Nevanlinna theories with vanishing and inﬁnite periods. Michigan Math. J. , 66(3):451–480, 2017

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Y. M. Chiang and S. N. M. Ruijsenaars. On the Nevanlinna or der of meromorphic solutions to linear analytic diﬀerence equations. Stud. Appl. Math. , 116(3):257–287, 2006

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2009

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Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

A new proximity function estimate on the quotient of the difference and the derivative of a meromorphic function

L. Asikainen, J. M. Huusko, and R. Korhonen. A new proximi ty function estimate on the quotient of the diﬀerence and the derivative of a meromorphi c function. arXiv preprint arXiv:2306.06729, 2023

work page Pith review arXiv 2023

[2] [2]

Bergweiler and J

W. Bergweiler and J. K. Langley. Zeros of diﬀerences of me romorphic functions. Math. Proc. Cambridge Philos. Soc. , 142(1):133–147, 2007

work page 2007

[3] [3]

Cherry and Z

W. Cherry and Z. Ye. Nevanlinna’s theory of value distribution . Springer Monogr. Math. Springer-Verlag, Berlin, 2001. The second main theorem and its error terms

work page 2001

[4] [4]

Cherry and Z

W. Cherry and Z. Ye. Nevanlinna’s theory of value distribution . Springer Science & Business Media, 2013

work page 2013

[5] [5]

Y. M. Chiang and S. J. Feng. On the growth of logarithmic di ﬀerences, diﬀerence quotients and logarithmic derivatives of meromorphic functions. Trans. Amer. Math. Soc., 361(7):3767– 3791, 2009

work page 2009

[6] [6]

Y. M. Chiang and X. D. Luo. Diﬀerence Nevanlinna theories with vanishing and inﬁnite periods. Michigan Math. J. , 66(3):451–480, 2017

work page 2017

[7] [7]

Y. M. Chiang and S. N. M. Ruijsenaars. On the Nevanlinna or der of meromorphic solutions to linear analytic diﬀerence equations. Stud. Appl. Math. , 116(3):257–287, 2006

work page 2006

[8] [8]

F. Gross. On the distribution of values of meromorphic fu nctions. Trans. Amer. Math. Soc. , 131:199–214, 1968

work page 1968