pith. sign in

arxiv: 1907.03601 · v1 · pith:CPYRQ6RUnew · submitted 2019-07-05 · 🧮 math.GM

Quantum Montgomery identity and some quantum integral inequalities

Mehmet Kunt , Artion Kashuri , Tingsong Du This is my paper

Pith reviewed 2026-05-25 01:41 UTC · model grok-4.3

classification 🧮 math.GM
keywords quantum Montgomery identityOstrowski inequalitiesquantum integral inequalitiesq-calculusMontgomery identityintegral inequalities
0
0 comments X

The pith

A quantum version of the Montgomery identity is established using quantum integral operators and applied to derive Ostrowski-type inequalities.

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

The paper introduces a quantum analogue of the Montgomery identity through quantum integral operators. Using this identity, it proves certain quantum integral inequalities of Ostrowski type. A sympathetic reader would care because the results extend classical integral inequalities into the quantum calculus setting, providing bounds on how much a function can deviate from its quantum average. The work also relates the new findings to results in earlier papers.

Core claim

The authors discover a new version of the celebrated Montgomery identity via quantum integral operators and establish certain quantum integral inequalities of Ostrowski type by using this identity, while considering relevant connections to earlier published papers.

What carries the argument

The quantum Montgomery identity, a representation of a function via quantum integrals that serves as the basis for the inequalities.

If this is right

  • Ostrowski-type bounds hold in the quantum integral setting.
  • The quantum inequalities provide estimates for the difference between a function value and its quantum integral average.
  • The new identity links to and generalizes results from prior work on similar integral inequalities.

Where Pith is reading between the lines

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

  • The identity might support error estimates in numerical methods adapted to quantum calculus.
  • Similar quantum versions could be derived for other classical identities in integral inequality theory.
  • The approach may extend to inequalities involving higher-order quantum derivatives if the basic case holds.

Load-bearing premise

The quantum integral operators permit a direct analogue of the classical Montgomery identity derivation without additional restrictions on the parameter q or the function class.

What would settle it

A concrete counterexample consisting of a specific continuous function and a fixed q value where the claimed quantum Montgomery identity fails to hold.

read the original abstract

We discover a new version of the celebrated Montgomery identity via quantum integral operators and establish certain quantum integral inequalities of Ostrowski type by using this identity. Relevant connections of the results obtained in this work with those deduced in earlier published papers are also considered.

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.

Referee Report

1 major / 2 minor

Summary. The paper claims to derive a new quantum analogue of the Montgomery identity using quantum integral operators (Jackson q-integral and q-derivative) and then applies this identity to obtain Ostrowski-type quantum integral inequalities. It also discusses connections to results in earlier papers on the topic.

Significance. If the central derivation is correct and the identity holds under the stated conditions, the work would extend classical Montgomery and Ostrowski results into q-calculus, a field with applications in quantum physics and special functions. The manuscript does not provide machine-checked proofs or parameter-free derivations.

major comments (1)
  1. [Derivation of the quantum Montgomery identity (likely §2-3)] The central claim that a direct quantum analogue of the Montgomery identity exists and reproduces the classical kernel without residual q-factors rests on an unverified assumption about the q-fundamental theorem and integration-by-parts. The skeptic's concern is load-bearing: if boundary terms in the q-integral definition (typically (1-q) sum q^k f(...)) introduce mismatches for 0<q<1, the subsequent Ostrowski inequalities do not follow. The paper must explicitly state the function class, range of q, and verify the identity step-by-step rather than assuming the classical derivation carries over verbatim.
minor comments (2)
  1. [Abstract] The abstract is too brief and does not indicate the precise quantum operators or any restrictions on q and f; this should be expanded for clarity.
  2. [Introduction or concluding section] Connections to earlier papers are mentioned but not compared in detail (e.g., which inequalities are new versus recovered); a table or explicit list would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on our manuscript. We address the major comment on the derivation of the quantum Montgomery identity below, providing clarifications and committing to revisions for improved explicitness.

read point-by-point responses

  1. Referee: [Derivation of the quantum Montgomery identity (likely §2-3)] The central claim that a direct quantum analogue of the Montgomery identity exists and reproduces the classical kernel without residual q-factors rests on an unverified assumption about the q-fundamental theorem and integration-by-parts. The skeptic's concern is load-bearing: if boundary terms in the q-integral definition (typically (1-q) sum q^k f(...)) introduce mismatches for 0<q<1, the subsequent Ostrowski inequalities do not follow. The paper must explicitly state the function class, range of q, and verify the identity step-by-step rather than assuming the classical derivation carries over verbatim.

    Authors: We appreciate the referee raising this important point about rigor in the q-calculus setting. The derivation in Section 2 applies the q-analogue of the fundamental theorem of calculus and the q-integration-by-parts formula directly to the Jackson q-integral, which is defined as the infinite sum (1-q) sum q^k f(...). This accounts for the boundary terms explicitly, yielding the stated quantum Montgomery identity with the classical kernel form (no residual q-factors) for the functions considered. The manuscript assumes q-differentiable functions on a closed interval [a,b] with 0<q<1. However, to address the concern about verification, we will revise the manuscript to include a fully expanded step-by-step proof of the identity, explicitly computing the boundary contributions to confirm they cancel appropriately, and stating the precise function class and q-range at the outset of Section 2. This will also clarify why the subsequent Ostrowski inequalities follow without issue. revision: yes

Circularity Check

0 steps flagged

Derivation of quantum Montgomery identity proceeds from operator definitions without reduction to inputs

full rationale

The paper states it discovers the quantum Montgomery identity via quantum integral operators and uses it to obtain Ostrowski-type inequalities. No equations or steps are shown that define the identity in terms of itself, rename a fitted quantity as a prediction, or rely on a self-citation chain for the core identity. The mention of connections to earlier papers is peripheral and does not carry the derivation. The claimed result is therefore independent of its inputs once the quantum operators and their fundamental theorem are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities; all fields left empty due to lack of detail.

pith-pipeline@v0.9.0 · 5550 in / 938 out tokens · 33267 ms · 2026-05-25T01:41:24.205963+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    A. A. Aljinovi´ c, Montgomery identity and Ostrowski typ e inequalities for Riemann-Liouville fractional integral, J. Math., Article ID 503195 (2014) 1-6

    work page 2014

  2. [2]

    G. A. Anastassiou, Ostrowski type inequalities, Proc. A mer. Math. Soc., 123 (12) (1995) 3775-3781

    work page 1995

  3. [3]

    M. H. Annaby, Z. S. Mansour, q- Fractional Calculus and Equations, Springer, Heidelberg , (2012)

    work page 2012

  4. [4]

    Alomari, M

    M. Alomari, M. Darus, S. S. Dragomir, P. Cerone, Ostrowsk i type inequalities for functions whose derivatives are s-convex in the second sense, Appl. Math. Lett., 23 (2010) 107 1-1076

    work page 2010

  5. [5]

    N. Alp, M. Z. Sarıkaya, A new definition and properties of q uantum integral which calls q-integral, Konuralp J. Math., 5 (2) (2017) 146-159

    work page 2017

  6. [6]

    N. Alp, M. Z. Sarıkaya , M. Kunt, ˙I. ˙I¸ scan,q-Hermite–Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and qua si-convex functions, J. King Saud Univ. Sci., 30 (2) (2018) 193-203

    work page 2018

  7. [7]

    Budak, M

    H. Budak, M. Z. Sarıkaya, On generalized Ostrowski-type inequalities for functions whose firrst derivatives absolute values are convex, Turkish J. Ma th., 40 (2016) 1193-1210

    work page 2016

  8. [8]

    Cerone, S

    P. Cerone, S. S. Dragomir, On some inequalities arising f rom montgomery’s identity (mont- gomery’s identity), J. Comput. Anal. Appl., 5 (4) (2003) 341 -367

    work page 2003

  9. [9]

    S. S. Dragomir, T. M. Rassias, Ostrowski type inequaliti es and applications in numerical integration, Kluwer Academic Publishers, 2002

    work page 2002

  10. [10]

    Farid, Some new Ostrowski type inequalities via frac tional integrals, Int

    G. Farid, Some new Ostrowski type inequalities via frac tional integrals, Int. J. Anal. Appl., 14 (2017) 64-68

    work page 2017

  11. [11]

    ˙I¸ scan, Ostrowski type inequalities for p-convex functions, New trends Math

    ˙I. ˙I¸ scan, Ostrowski type inequalities for p-convex functions, New trends Math. Sci., 4(3) (2016) 140-150

    work page 2016

  12. [12]

    V. Kac, P. Cheung: Quantum calculus, Springer (2001)

    work page 2001

  13. [13]

    Kavurmacı, M

    H. Kavurmacı, M. E. ¨Ozdemir, M. Avcı, New Ostrowski type inequalities for m-convex func- tions and applications, Hacet. J. Math. Stat., 40 (2) (2011) 135-145

    work page 2011

  14. [14]

    M. E. Kiri¸ s, M. Z. Sarıkaya, On Ostrowski type inequali ties and ˇCebyˇ sev type inequalities with applications, Filomat, 29 (8) (2015) 1695-1713

    work page 2015

  15. [15]

    U. S. Kirmaci, Inequalities for differentiable mapping s and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 1 47 (2004) 137-146

    work page 2004

  16. [16]

    Kunt , ˙I

    M. Kunt , ˙I. ˙I¸ scan, N. Alp, M. Z. Sarıkaya, (p, q)-Hermite–Hadamard inequalities and ( p, q)- estimates for midpoint type inequalities via convex and qua si-convex functions, Rev. R. Acad. Cienc. Exactas F ´ ıs. Nat. Ser. A Math., 112 (2018) 969-992

    work page 2018

  17. [17]

    M. Kunt, M. A. Latif, ˙I. ˙I¸ scan, S. S. Dragomir, Quantum Hermite-Hadamard type ineq uality and some estimates of quantum midpoint type inequalities fo r double integrals, Sigma J. Eng. Nat. Sci., 37 (1) (2019) 207-223

    work page 2019

  18. [18]

    W. J. Liu, H. F. Zhuang, Some quantum estimates of Hermit e–Hadamard inequalities for convex functions, J. Appl. Anal. Comput., 7 (2) (2017) 501–5 22

    work page 2017

  19. [19]

    Liu, Some Ostrowski type inequalities, Math

    Z. Liu, Some Ostrowski type inequalities, Math. Comput . Model., 48 (2008) 949-960

    work page 2008

  20. [20]

    Mat/suppress loka, Ostrowski type inequalities for functions whose derivatives are h-convex via fractional integrals, J

    M. Mat/suppress loka, Ostrowski type inequalities for functions whose derivatives are h-convex via fractional integrals, J. Sci. Res. & Rep., 3(12) (2014) 1633 -1641

    work page 2014

  21. [21]

    D. S. Mitrinovi´ c, J. E. Peˇ cari´ c, and A. M. Fink, Inequalities for functions and their integrals and derivatives, Kluwer Academic, Dordrecht, 1991

    work page 1991

  22. [22]

    M. A. Noor, M. U. Awan, K. I. Noor, Quantum Ostrowski ineq ualities for q-differentiable convex functions, J. Math. Inequal., 10 (4) (2016) 1013-101 8

    work page 2016

  23. [23]

    M. A. Noor, K. I. Noor, M. U. Awan, Some quantum estimates for Hermite-Hadamard inequalities, Appl. Math. Comput. 251 (2015) 675–679

    work page 2015

  24. [24]

    M. A. Noor, K. I. Noor, M. U. Awan, Some quantum integral i nequalities via preinvex functions, Appl. Math. Comput. 269 (2015) 242–251

    work page 2015

  25. [25]

    Ostrowski, ¨Uber die Absolutabweichung einer differentienbaren Funkti onen von ihren In- tegralmittelwert

    A. Ostrowski, ¨Uber die Absolutabweichung einer differentienbaren Funkti onen von ihren In- tegralmittelwert. Comment. Math. Hel, 10 (1938), 226–227

    work page 1938

  26. [26]

    M. E. ¨Ozdemir, H. Kavurmacı, M. Avcı, Ostrowski type inequalitie s for convex functions, Tamkang J. Math., 45 (4) (2014) 335-340

    work page 2014

  27. [27]

    M. Z. Sarıkaya, H. Budak, Generalized Ostrowski type in equalities for local fractional inte- grals, Proc. Amer. Math. Soc., 145(4) (2017) 1527-1538. 16 M. KUNT, A. KASHURI, T.-S. DU

    work page 2017

  28. [28]

    E. Set, M. E. ¨Ozdemir, M. Z. Sarıkaya, New inequalities of Ostrowski’s ty pe for s-convex functions in the second sense with applications, Facta Univ . Ser. Math. Inform., 27 (1) (2012) 67-82

    work page 2012

  29. [29]

    Sudsutad, S

    W. Sudsutad, S. K. Ntouyas, J. Tariboon, Quantum integr al inequalities for convex functions, J. Math. Inequal., 9 (3) (2015) 781–793

    work page 2015

  30. [30]

    Tariboon, S

    J. Tariboon, S. K. Ntouyas, Quantum calculus on finite in tervals and applications to impulsive difference equations, Adv. Difference Equ. 282 (2013) 1-19

    work page 2013

  31. [31]

    Tariboon, S

    J. Tariboon, S. K. Ntouyas, Quantum integral inequalit ies on finite intervals, J. Inequal. Appl. Article ID 121 (2014) 1-13

    work page 2014

  32. [32]

    Tun¸ c, E

    M. Tun¸ c, E. G¨ ov, S. Balge¸ cti, Simpson type quantum integral inequalities for convex func- tions, Miskolc Math. Notes, 19 (1) (2018) 649-664

    work page 2018

  33. [33]

    Zhang , T.-S

    Y. Zhang , T.-S. Du, H. W ang , Y.-J. Shen, Different types o f quantum integral inequalities via ( α, m)-convexity, J. Inequal. Appl., Article ID 264 (2018) 1-24

    work page 2018

  34. [34]

    H. F. Zhuang, W. J. Liu, J. Park, Some quantum estimates o f Hermite-Hadamard inequalities for quasi-convex functions, Mathematics, Article ID 152, 7 (2) (2019) 1-18. 1Department of Mathematics, F aculty of Sciences, Karadeniz Technical University, 61080, Trabzon, Turkey E-mail address : mkunt@ktu.edu.tr 2Department of Mathematics, F aculty of Technical S...

    work page 2019