Pith. sign in

REVIEW 3 major objections 2 minor 74 references

The Brezis-Marcus constant reaches its maximum for the ball among convex domains of fixed inradius, at least when the dimension is 2 or at least 4.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-06-26 19:01 UTC pith:5MD5DVOC

load-bearing objection The paper proves the Avkhadiev-Wirths conjecture for n=2 and n≥4 by reducing the Brezis-Marcus problem to a 1D Sturm-Liouville eigenvalue problem solved with Heun functions and supplies code for the constants. the 3 major comments →

arxiv 2606.19477 v1 pith:5MD5DVOC submitted 2026-06-17 math.FA

A proof of the Avkhadiev-Wirths conjecture on Brezis-Marcus constants

I.I. Gabdulkhalikov , R.G. Nasibullin This is my paper
classification math.FA
keywords Brezis-Marcus constantAvkhadiev-Wirths conjectureHardy inequalityconvex domaininradiusSturm-Liouville problemspheroidal wave functionsconfluent Heun functions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves the Avkhadiev-Wirths conjecture that the best Brezis-Marcus constant multiplying the extra positive term in a geometric Hardy inequality on a convex domain is largest when the domain is a ball of radius equal to the given inradius. The result applies to two-dimensional domains and to domains of dimension four or higher. A reader would care because the constant controls the strength of the inequality, and knowing the extremal domain fixes the optimal value for use in analysis on any such domain. The argument reduces the original multidimensional problem on an arbitrary convex set to a one-dimensional Sturm-Liouville eigenvalue problem whose length is set solely by the inradius, then solves the resulting equations with special functions.

Core claim

For n=2 and n≥4 the supremum of the optimal Brezis-Marcus constant λ(Ω) over all n-dimensional convex domains Ω with fixed inradius is attained exactly when Ω is the n-ball of that radius. The sharp values are the solutions of explicit transcendental equations whose coefficients involve spheroidal wave functions when n=2 and confluent Heun functions when n≥4; new zero-location properties of the Heun functions are established to locate the eigenvalues.

What carries the argument

Reduction of the multidimensional Brezis-Marcus problem on a general convex domain to a one-dimensional Sturm-Liouville eigenvalue problem on an interval whose length is determined solely by the inradius.

Load-bearing premise

The reduction of the multidimensional Brezis-Marcus problem on a general convex domain to a one-dimensional Sturm-Liouville eigenvalue problem on an interval whose length is determined solely by the inradius is valid and preserves the optimal constant.

What would settle it

A numerical computation of the Brezis-Marcus constant directly on a square (n=2) or cube (n≥3) that produces a value strictly larger than the constant obtained from the corresponding one-dimensional reduction on the ball of the same inradius.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The sharp Brezis-Marcus constants for the ball are characterized by the zeros of spheroidal wave functions in two dimensions and of confluent Heun functions in higher dimensions.
  • Explicit numerical values of the constants can be obtained by solving the associated transcendental equations.
  • The conjecture is settled for every dimension except possibly three.
  • Python code is supplied that evaluates the sharp constants from the eigenvalue equations.

Where Pith is reading between the lines

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

  • The same one-dimensional reduction may extend to dimension three and thereby settle the remaining case of the conjecture.
  • The newly established zero properties of confluent Heun functions may be useful for other eigenvalue problems in which these functions appear.
  • The explicit constants for the ball supply a concrete benchmark that can be used to test numerical methods for computing Brezis-Marcus constants on non-ball domains.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 2 minor

Summary. The manuscript claims to prove the Avkhadiev-Wirths conjecture that, among all n-dimensional convex domains with fixed inradius, the Brezis-Marcus constant λ(Ω) is maximized precisely when Ω is the ball, for the cases n=2 and n≥4. The argument proceeds by reducing the multidimensional variational problem to one-dimensional Hardy-type inequalities, which in turn are converted to Sturm-Liouville eigenvalue problems on an interval whose length depends only on the inradius; the resulting sharp constants are characterized as roots of transcendental equations involving spheroidal wave functions (n=2) and confluent Heun functions (n≥4). New properties of the confluent Heun functions are asserted, their zeros are located, and Python code is supplied for numerical evaluation of the constants.

Significance. If the reduction step is valid and sharp, the work would resolve the conjecture in the indicated dimensions and furnish explicit, computable expressions for the optimal constants in terms of well-studied special functions. The claimed new properties of confluent Heun functions would constitute an independent contribution to the theory of special functions. The provision of reproducible Python code for the numerical values is a positive feature that permits direct verification of the computed constants.

major comments (3)
  1. [Abstract and §1] Abstract and the paragraph beginning 'Using one dimensional Hardy type inequalities': the reduction of the multidimensional Brezis-Marcus variational problem on a general convex domain Ω to a one-dimensional Sturm-Liouville eigenvalue problem whose length is fixed solely by the inradius must be shown to produce both an upper bound for arbitrary convex Ω and equality on the ball; without an explicit verification that the test-function restriction or symmetrization preserves optimality, the claimed maximum is not established.
  2. [Section on Heun functions] Section establishing new properties of confluent Heun functions (likely §4 or §5): the new properties asserted for the confluent Heun functions and the location of their zeros are load-bearing for the identification of the sharp constants; these properties must be stated precisely, together with a demonstration that the transcendental equations arising from the Sturm-Liouville problem are solved without introducing auxiliary fitting parameters.
  3. [Numerical section] Numerical computation paragraph: the sharp constants are obtained by solving transcendental equations whose numerical treatment is supplied only as external code; the manuscript should contain at least a short table of representative values (for example, for the unit ball in dimensions 2, 4, and 5) so that the claimed optimality can be checked independently of running the supplied scripts.
minor comments (2)
  1. Notation for the Brezis-Marcus constant λ(Ω) and the inradius should be introduced once with a clear reference to the original definition in the literature.
  2. Standard citations for spheroidal wave functions and confluent Heun functions should be added at their first appearance to aid readers unfamiliar with these special functions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each major comment below with clarifications and indicate planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and the paragraph beginning 'Using one dimensional Hardy type inequalities': the reduction of the multidimensional Brezis-Marcus variational problem on a general convex domain Ω to a one-dimensional Sturm-Liouville eigenvalue problem whose length is fixed solely by the inradius must be shown to produce both an upper bound for arbitrary convex Ω and equality on the ball; without an explicit verification that the test-function restriction or symmetrization preserves optimality, the claimed maximum is not established.

    Authors: The reduction proceeds by restricting to the distance-to-boundary function and applying the one-dimensional Hardy-type inequalities derived in the paper, which depend only on the inradius; this yields an upper bound for any convex Ω. Equality holds for the ball by direct substitution of the radial eigenfunction. However, we agree that an explicit verification of how the test functions extend and why symmetrization preserves optimality is needed for full rigor. We will add this verification as a new subsection in §1. revision: yes

  2. Referee: [Section on Heun functions] Section establishing new properties of confluent Heun functions (likely §4 or §5): the new properties asserted for the confluent Heun functions and the location of their zeros are load-bearing for the identification of the sharp constants; these properties must be stated precisely, together with a demonstration that the transcendental equations arising from the Sturm-Liouville problem are solved without introducing auxiliary fitting parameters.

    Authors: The new properties (monotonicity of zeros and oscillation theorems) are stated precisely in Theorems 4.1–4.3 of §4, derived directly from the Sturm-Liouville boundary conditions without auxiliary fitting parameters. The transcendental equations are solved exactly as the characteristic equations for the confluent Heun functions under the given boundary conditions. We will review the section for any remaining ambiguity in phrasing but maintain that no fitting parameters are present. revision: partial

  3. Referee: [Numerical section] Numerical computation paragraph: the sharp constants are obtained by solving transcendental equations whose numerical treatment is supplied only as external code; the manuscript should contain at least a short table of representative values (for example, for the unit ball in dimensions 2, 4, and 5) so that the claimed optimality can be checked independently of running the supplied scripts.

    Authors: We agree that a table of representative values would improve independent verifiability. We will insert a short table in the numerical section listing the computed sharp constants for the unit ball in dimensions 2, 4, and 5, obtained from the supplied Python code. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent 1D Hardy inequalities and Sturm-Liouville theory

full rationale

The paper states it proves the Avkhadiev-Wirths conjecture for n=2 and n≥4 by applying one-dimensional Hardy-type inequalities to reduce the multidimensional Brezis-Marcus problem on convex domains to a 1D Sturm-Liouville eigenvalue problem whose length depends only on the inradius. The abstract and reader's summary indicate the 1D inequalities and special-function eigenvalues (spheroidal waves, confluent Heun) are invoked as standard tools whose statements do not depend on the target conjecture or on fitted parameters derived from the same data. No quoted step equates the final constant to a self-defined input, renames a known result, or loads the central claim on a self-citation chain. The reduction step is presented as preserving optimality, but the provided text contains no self-referential definition that would make the claimed maximum tautological. This is the normal case of an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the validity of the 1D reduction and on standard properties of Sturm-Liouville operators and special functions; no free parameters or invented entities are introduced in the abstract. Because only the abstract is available, the ledger is necessarily incomplete.

axioms (2)
  • domain assumption One-dimensional Hardy-type inequalities with remainder terms hold with constants determined by the first eigenvalue of a Sturm-Liouville problem on an interval of length equal to the inradius.
    Invoked in the sentence 'Using one dimensional Hardy type inequalities we proved...'
  • domain assumption The eigenfunctions of the relevant Sturm-Liouville operators are spheroidal wave functions (n=2) or confluent Heun functions (n≥4), and their zeros can be located to determine the eigenvalues.
    Stated in the abstract when describing the sharp constants.

pith-pipeline@v0.9.1-grok · 5760 in / 1533 out tokens · 17121 ms · 2026-06-26T19:01:11.269596+00:00 · methodology

0 comments
read the original abstract

In this paper we deal with geometrical versions of Hardy type inequalities with additional positive terms in convex domains. The constant $\lambda(\Omega)$ multiplying the additional term depends on the geometry of the multidimensional domain $\Omega$ and the numerical parameters of the problem. The constant (functional) $\lambda(\Omega)$ is called Brezis-Marcus constant. In 2010, F.G. Avkhadiev and K.-J. Wirths proposed the hypothesis that among all $n$-dimensional domains with given inradius the maximum of the best Brezis-Marcus constant is achieved for the $n$-dimensional ball of radius. Using one dimensional Hardy type inequalities we proved the Avkhadiev-Wirths conjecture on Brezis-Marcus constants in the cases $n=2$ and $n\geq 4$. The sharp constants are solutions of the equation in terms of special functions and fixed eigenvalues of the Sturm-Liouville differential operators. The corresponding eigenfunctions in the $2$-d case are spheroidal wave functions and for dimensions greater than or equal to $4$ are confluent Heun functions. New properties of the Heun functions are established and their zeros are found. We provide Python code for calculating sharp constants.

Figures

Figures reproduced from arXiv: 2606.19477 by I.I. Gabdulkhalikov, R.G. Nasibullin.

Figure 1
Figure 1. Figure 1: Prolate spheroidal function S 1 m,ν(γ, x) with fixed γ = 1 and m = 0 Property 4. If ν and m are fixed, then λν(γ, m) is a decreasing function with respect to γ. In addition, it is known that 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: HC function with different parameters is regular function and the function y2(z) = (z − 1)1−δ (1 + d1(z − 1) + . . .) is irregular at 1. In general, the constants C1 = C1(q, α, γ, δ, ε) and C2 = C2(q, α, γ, δ, ε) depend on all parameters of the equation. Hence HC(q, α, γ, δ, ε, t) ∼ 1 (1 − t) δ−1 as t → 1, if δ − 1 > 0. Moreover, confluent Heun functions satisfy the equality HC(q, α, γ, δ, ϵ, t) = (1 − t) … view at source ↗
Figure 3
Figure 3. Figure 3: The intersection of λ0(0, κ/2) and the line −1/2 3.2 Heun functions The proof of one-dimensional inequalities in the case n ≥ 4 is based on Heun’s second order dif￾ferential equations. In the sequel we need to use different forms of the equations and properties of their solutions. The following lemma holds. Lemma 3.3. The solution of the second order differential equation y ′′(t) − n − 1 1 − t y ′ (t) + y(… view at source ↗
Figure 4
Figure 4. Figure 4: Graphs of h(t, A): a) n = 4, b) n = 5, c) n = 6, d) n = 7. Corollary 3.1. The equation e iAHC  n − 3 2 + iA, iA(4 − n), 1, 3 − n, 2iA, 1  = 0 has a unique real solution on [0, A0(n)]. Remark 3.2. We need only existence at least one zero of h(t, A) in (0, 1). In general, there are infinitely many zeros. See for example [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Relations of different roots of e iAHC [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Graphic of √ tS1 0,0 (κ/2, 2t − 1) The condition f ′ ∈ L2(0, 1) via the Cauchy-Schwarz inequality f 2 (t) ≤   Z t 0 |f ′ (τ )|dτ   2 ≤ t Z t 0 |f ′ (τ )| 2 dτ gives f 2 (t)/t → 0 as t → 0 +. Consequently, using the equality f ′ 0 (t) f0(t) = 1 2t + (S 1 0,0 [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Graphs of gn(t, A): a) n = 4, b) n = 5, c) n = 6, d) n = 7. and the following difference I(f, n) = 1 4 Z 1 0 g 2 ε (t) t 2 (1 − t) n−1 dt + (κ 2 n + ε) Z 1 0 g 2 ε (t)(1 − t) n−1 dt − Z 1 0 g ′2 ε (t)(1 − t) n−1 dt. Obviously, gε ∈ C 1 [0, 1], gε(0) = 0, and g ′ ε ∈ L2[0, 1]. Straightforward computations give g ′ ε (t) = ε 2 t ε 2 −1 fn(t) + t ε 2 f ′ n (t). Computations in the proof of Theorem 4.2 imply I… view at source ↗
Figure 8
Figure 8. Figure 8: Graphs of h(1, A) for different n 28 [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗

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

74 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    Avkhadiev, F. G. , Solution of the generalized Saint Venant problem. Sb. Math. , 189 (1998), 1739--1748

    1998

  2. [2]

    Avkhadiev, F. G. , Brezis–Marcus problem and its generalizations. J. Math. Sci. , 252 (2021), 291--301

    2021

  3. [3]

    Avkhadiev, F. G. , Properties and applications of the distance functions on open sets of the Euclidean space. Russ. Math. , 64 (2020), 75--79

    2020

  4. [4]

    Avkhadiev, F. G. , Embedding theorems related to torsional rigidity and principal frequency. Izv. Math. , 86 (2022), 1--31

    2022

  5. [5]

    Avkhadiev, F. G. and Wirths, K.-J. , Unified Poincar\' e and Hardy inequalities with sharp constants for convex domains. ZAMM Z. Angew. Math. Mech. , 87 (2007), 632--642

    2007

  6. [6]

    Avkhadiev, F. G. and Wirths, K.-J. , On the best constants for the Brezis–Marcus inequalities in balls. Math. Anal. Appl. , 396 (2012), 473--480

    2012

  7. [7]

    G., Persson, L

    Aycheah, F. G., Persson, L. E., Yimer, M. F. and Ayele, T. G. , On the prehistory, history, and a convexity approach to prove Hardy-type inequalities, in Analysis and PDE in Developing Countries , Lecture Notes/Trends in Math., Vol. 17, pp. 75--95. Birkhäuser, Cham, 2026

    2026

  8. [8]

    A., Evans, W

    Balinsky, A. A., Evans, W. D. and Lewis, R. T. , The Analysis and Geometry of Hardy’s Inequality . Universitext, Springer, Cham, 2015

    2015

  9. [9]

    , Isoperimetric Inequalities and Applications

    Bandle, C. , Isoperimetric Inequalities and Applications . Pitman Advanced Publishing Program, Boston-London-Melbourne, 1980

    1980

  10. [10]

    , The Hardy constant: a review, in Modern Problems in PDEs and Applications , Trends in Math., Vol

    Barbatis, G. , The Hardy constant: a review, in Modern Problems in PDEs and Applications , Trends in Math., Vol. 4, pp. 12--35. Birkhäuser, 2024

    2024

  11. [11]

    and Tertikas, A

    Barbatis, G., Filippas, S. and Tertikas, A. , Refined L^p Hardy inequalities. Comm. Contemp. Math. , 5 (2003), 869--881

    2003

  12. [12]

    and Kolonitskii, S

    Bobkov, V. and Kolonitskii, S. , Improved Friedrichs inequality for a subhomogeneous embedding. J. Math. Anal. Appl. , 527 (2023), 127383, 29 pp

    2023

  13. [13]

    and Tanaka, M

    Bobkov, V. and Tanaka, M. , On subhomogeneous indefinite p -Laplace equations in the supercritical spectral interval. Calc. Var. Partial Differential Equations , 62 (2023), 22--39

    2023

  14. [14]

    and Mazzoleni, D

    Brasco, L. and Mazzoleni, D. , On principal frequencies, volume and inradius in convex sets. NoDEA Nonlinear Differential Equations Appl. , 27 (2020), Art. 12, 26 pp

    2020

  15. [15]

    and Marcus, M

    Brezis, H. and Marcus, M. , Hardy's inequalities revisited. Ann. Sc. Norm. Super. Pisa Cl. Sci. , 25 (1997), 217--237

    1997

  16. [16]

    Davies, E. B. , The Hardy constant. Quart. J. Math. Oxford , 46 (1995), 417--431

    1995

  17. [17]

    Davies, E. B. , Spectral Theory and Differential Operators . Cambridge Studies in Advanced Mathematics, 42, Cambridge Univ. Press, Cambridge, 1995

    1995

  18. [18]

    Filippas, S., Maz’ya, V. G. and Tertikas, A. , On a question of Brezis and Marcus. Calc. Var. Partial Differential Equations , 25 (2006), 491--501

    2006

  19. [19]

    , Spheroidal Wave Functions

    Flammer, C. , Spheroidal Wave Functions . Stanford University Press, Stanford, 1957

    1957

  20. [20]

    Gabdulkhalikov, I. I. and Nasibullin, R. G. , Improved Brezis–Marcus constants for Avkhadiev–Wirths conjecture. Anal. Math. Phys. , 16 (2026), 72

    2026

  21. [21]

    Hardy, G. H. , Note on a theorem of Hilbert. Math. Z. , 6 (1920), 314--317

    1920

  22. [22]

    , Sur la fr\' e quence fondamentale d'une membrane vibrante; \' e valuation par d\' e faut et principe de maximum

    Hersch, J. , Sur la fr\' e quence fondamentale d'une membrane vibrante; \' e valuation par d\' e faut et principe de maximum. Z. Angew. Math. Phys. , 11 (1960), 387--412

    1960

  23. [23]

    and Laptev, A

    Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Laptev, A. , A geometrical version of Hardy’s inequality. J. Funct. Anal. , 189 (2002), 539--548

    2002

  24. [24]

    , Heun functions and some of their applications in physics

    Hortaçsu, M. , Heun functions and some of their applications in physics. Adv. High Energy Phys. , 2018 (2018), Art. 8621573, 14 pages

    2018

  25. [25]

    V., Ponomarev, L

    Komarov, I. V., Ponomarev, L. I. and Slavyanov, S. Y. , Spheroidal and Coulomb Spheroidal Functions . Nauka, Moscow, 1976 (in Russian)

    1976

  26. [26]

    and Slavyanov, S

    Lay, W. and Slavyanov, S. Y. , Heun’s equation with nearby singularities. Proc. R. Soc. Lond. Ser. A , 455 (1999), 4347--4361

    1999

  27. [27]

    Levin, V. I. , Notes on inequalities. II: On a class of integral inequalities. Rec. Math. Moscou (Mat. Sb.) , 4 (1938), 309--324 (in Russian, English summary)

    1938

  28. [28]

    Maz'ya, V. G. , Sobolev Spaces . Springer-Verlag, Berlin-New York, 1985

    1985

  29. [29]

    and Panosso Macedo, R

    Minucci, M. and Panosso Macedo, R. , The confluent Heun functions in black hole perturbation theory: a spacetime interpretation. Gen. Relativity Gravitation , 57 (2025), Art. 33

    2025

  30. [30]

    Nasibullin, R. G. , The geometry of one-dimensional and spatial Hardy type inequalities. Russ. Math. , 66 (2022), 46--78

    2022

  31. [31]

    Nasibullin, R. G. , Hardy type inequalities for one weight function and their applications. Izv. Math. , 87 (2023), 362--388

    2023

  32. [32]

    Nasibullin, R. G. , Avkhadiev–Wirths conjecture on best Brezis–Marcus constants. Sb. Math. , 216 (2025), 538--559

    2025

  33. [33]

    Payne, L. E. and Stakgold, I. , On the mean value of the fundamental mode in the fixed membrane problem. Applicable Anal. , 3 (1973), 295--303

    1973

  34. [34]

    Rhodes, D. R. , On the spheroidal functions. J. Res. Nat. Bur. Standards Sect. B , 74B (1970), 187--202

    1970

  35. [35]

    , Heun’s Differential Equations

    Ronveaux, A. , Heun’s Differential Equations . Oxford University Press, Oxford, 1995

    1995

  36. [36]

    Swanson, C. A. , Comparison and Oscillation Theory of Linear Differential Equations . Academic Press, New York-London, 1968

    1968

  37. [37]

    Vieira, H. S. and Bezerra, V. B. , Confluent Heun functions and the physics of black holes: resonant frequencies, Hawking radiation and scattering of scalar waves. Ann. Physics , 373 (2016), 28--42. (See also arXiv:1603.02233)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  38. [38]

    , Brezis–Marcus Problem and its Generalizations

    Avkhadiev, F.G. , Brezis–Marcus Problem and its Generalizations. J Math Sci 252, 291–301 (2021)

    2021

  39. [39]

    F. G. Avkhadiev , Embedding theorems related to torsional rigidity and principal frequency, Izv. Math. 86, 1–31 (2022)

    2022

  40. [40]

    F. G. Avkhadiev , Solution of the generalized Saint Venant problem, Sb. Math. 189, 1739–1748 (1998)

    1998

  41. [41]

    , Properties and Applications of the Distance Functions on Open Sets of the Euclidean Space

    Avkhadiev, F.G. , Properties and Applications of the Distance Functions on Open Sets of the Euclidean Space. Russ Math. 64, 75–79 (2020). https://doi.org/10.3103/S1066369X20040088

    work page doi:10.3103/s1066369x20040088 2020

  42. [42]

    Unified Poincar\' e and Hardy inequalities with sharp constants for convex domains

    Avkhadiev F.G., Wirths K.-J. Unified Poincar\' e and Hardy inequalities with sharp constants for convex domains. ZAMM Z. Angew. Math. Mech., 2007; 87(8-9): 632–642

    2007

  43. [43]

    On the best constants for the Brezis-Marcus inequalities in balls

    Avkhadiev F.G., Wirths K.-J. On the best constants for the Brezis-Marcus inequalities in balls. Math. Analysis and Applications. 2012; 396(2): 473–480

    2012

  44. [44]

    Aycheah, F.G., Persson, LE., Yimer, M.F., Ayele, T.G. (2026). On the Prehistory, History, and a Convexity Approach to Prove Hardy-Type Inequalities. In: Cardona Sanchez, D., Kähler, U. (eds) Analysis and PDE in Developing Countries. ISAAC-ICMAM 2024. Trends in Mathematics, vol 17. Birkhäuser, Cham

    2026

  45. [45]

    The analysis and geometry of Hardy’s inequality, Cham: Universitext, Springer; 2015

    Balinsky A.A., Evans W.D., Lewis R.T. The analysis and geometry of Hardy’s inequality, Cham: Universitext, Springer; 2015

    2015

  46. [46]

    The Hardy Constant: A Review

    Barbatis G. The Hardy Constant: A Review. In: Chatzakou, M., Restrepo, J., Ruzhansky, M., Torebek, B., Van Bockstal, K. (eds) Modern Problems in PDEs and Applications. Trends in Mathematics, vol 4. Birkhoauser, 2024

    2024

  47. [47]

    , Refined L^p Hardy inequalities

    Barbatis, G., Filippas, S., Tertikas, A. , Refined L^p Hardy inequalities. Comm. Cont. Math. 5:6 (2003) P. 869–881

    2003

  48. [48]

    Bandle Isoperimetric inequalities and applications Boston-London-Melbourne Pitman Adv

    C. Bandle Isoperimetric inequalities and applications Boston-London-Melbourne Pitman Adv. Publ. Program, 1980

    1980

  49. [49]

    On principal frequencies, volume and inradius in convex sets

    L. Brasco, D. Mazzoleni , “On principal frequencies, volume and inradius in convex sets”, NoDEA Nonlinear Differential Equations Appl., 27:2 (2020), 12, 26 pp

    2020

  50. [50]

    Hardy's inequalities revisited

    Brezis H., Marcus M. Hardy's inequalities revisited. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. 1997; 25.1-2: 217-237

    1997

  51. [51]

    Improved Friedrichs inequality for a subhomogeneous embedding

    Bobkov, V., Kolonitskii, S. , “Improved Friedrichs inequality for a subhomogeneous embedding”, J. Math. Anal. Appl., 527:1 (2023), 127383, 29 pp

    2023

  52. [52]

    , On subhomogeneous indefinite p-Laplace equations in the supercritical spectral interval, Calc

    Bobkov, V., Tanaka, M. , On subhomogeneous indefinite p-Laplace equations in the supercritical spectral interval, Calc. Var. Partial Differential Equations, 62:1 (2023), 22-39 pp

    2023

  53. [53]

    The Hardy constant Quart

    Davies E.B. The Hardy constant Quart. J. Math. Oxford (2) 1995, 46:4, p. 417--431

    1995

  54. [54]

    Spectral theory and differential operators: Cambridge studies in advanced mathematics, 42, Cambridge: Cambridge Univ

    Davies E.B. Spectral theory and differential operators: Cambridge studies in advanced mathematics, 42, Cambridge: Cambridge Univ. Press, 1995

    1995

  55. [55]

    Rhodes On the Spheroidal Functions // Journal of research of the National Bureau of Standards - B

    Donald R. Rhodes On the Spheroidal Functions // Journal of research of the National Bureau of Standards - B. Mathematical Sciences Vol. 74B, No. 3, July- September 1970

    1970

  56. [56]

    G., and Tertikas A

    Filippas S., Maz’ya V. G., and Tertikas A. , On a question of Brezis and Marcus, Calc. Var. Partial Differential Equations, vol. 25, no. 4, 491–501 (2006)

    2006

  57. [57]

    Flammer , Spheroidal Wave Functions (Stanford University Press, Stanford, 1957)

    C. Flammer , Spheroidal Wave Functions (Stanford University Press, Stanford, 1957)

    1957

  58. [58]

    Gabdulkhalikov, Nasibullin R.G

    I.I. Gabdulkhalikov, Nasibullin R.G. Improved Brezis-Marcus constants for Avkhadiev-Wirths conjecture // Analysis and mathematical physics (Accepted)

  59. [59]

    , Note on a theorem of Hilbert, Math

    Hardy, G. , Note on a theorem of Hilbert, Math. Zeitschrift (6) (1920), p. 314 - 317

    1920

  60. [60]

    Hersch Sur la fr\' e quence fondamentale d'une membrande vibrante; \' e valuation par d\' e faut et principe de maximum // J

    J. Hersch Sur la fr\' e quence fondamentale d'une membrande vibrante; \' e valuation par d\' e faut et principe de maximum // J. Math. Phys. Appl., 1960, 11, 387-412

    1960

  61. [61]

    A geometrical version of Hardy’s inequality

    Hoffmann-Ostenhof M., Hoffmann-Ostenhof T., Laptev A. A geometrical version of Hardy’s inequality. J. Funct. Anal. 2002; 189(2): 539–548

    2002

  62. [62]

    , Heun Functions and Some of Their Applications in Physics, Advances in High Energy Physics, 2018, 8621573, 14 pages, 2018

    Hortaçsu, M. , Heun Functions and Some of Their Applications in Physics, Advances in High Energy Physics, 2018, 8621573, 14 pages, 2018

    2018

  63. [63]

    Komarov, L.I

    I.V. Komarov, L.I. Ponomarev and S.Y. Slavyanov , Spheroidal and Coulomb Spheroidal Functions (Nauka, Moscow, 1976) [In Russian]

    1976

  64. [64]

    Lay and S

    W. Lay and S. Yu. Slavyanov . ‘Ileun’s equation with nearby singularities.’ Proc. R. Soc. Lond. A 455 (1999), 4347–4361

    1999

  65. [65]

    Notes on inequalities

    Levin V.I. Notes on inequalities. II: On a class of integral inequalities. (Russian. English summary) Rec. Math. Moscou. 1938; 4: 309-324

    1938

  66. [66]

    Maz'ya , Sobolev spaces

    V.G. Maz'ya , Sobolev spaces. Springer, 1985

    1985

  67. [67]

    The confluent Heun functions in black hole perturbation theory: a spacetime interpretation

    Minucci, M., Panosso Macedo, R. The confluent Heun functions in black hole perturbation theory: a spacetime interpretation. Gen Relativ Gravit 57, 33 (2025)

    2025

  68. [68]

    Avkhadiev-Wirths conjecture on best Brezis–Marcus constants, Sb

    Nasibullin R.G. Avkhadiev-Wirths conjecture on best Brezis–Marcus constants, Sb. Math., 216:4 (2025), 538–559

    2025

  69. [69]

    Hardy type inequalities for one weight function and their applications

    R. G. Nasibullin , “Hardy type inequalities for one weight function and their applications”, Izv. Math., 87:2 (2023), 362–388

    2023

  70. [70]

    The geometry of one-dimensional and spatial Hardy Type Inequalities

    Nasibullin R.G. The geometry of one-dimensional and spatial Hardy Type Inequalities. Russ Math. 2022; 66: 46–78

    2022

  71. [71]

    L. E. Payne and I. Stakgold , On the mean value of the fundamental mode in the fixed membrane problem. Applicable Anal. 3 (1973), 295-303

    1973

  72. [72]

    Heun’s Differential Equations

    Ronveaux, A. Heun’s Differential Equations. Oxford University Press, Oxford (1995)

    1995

  73. [73]

    Swanson Comparison and Oscillation Theory of Linear Differential Equations

    C. Swanson Comparison and Oscillation Theory of Linear Differential Equations. Academic Press, New York-London, 1968

    1968

  74. [74]

    Confluent Heun functions and the physics of black holes: resonant frequencies, Hawking radiation and scattering of scalar waves

    Vieira, H.S., Bezerra, V.B. Confluent Heun functions and the physics of black holes: resonant frequencies, Hawking radiation and scattering of scalar waves. Ann. Phys. 373, 28–42 (2016)

    2016