The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H C Yang

Ashbaugh, Mark S.

doi:10.1007/bf02829638

Cited by 73 publications

(76 citation statements)

References 20 publications

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“…We also prove that both of Yang's inequalities (see [2], [3]) are stronger than both the PPW and HP results, thus supplying an argument left aside in [35]. An alternative proof of this latter result appears in [3] but our proof here is simpler and more direct.…”

Section: Introductionmentioning

confidence: 54%

“…Yang's derivation [35] is more circuitous, and in particular does not make explicit use of this crucial element of our proof. A full explanation of the method appeared first in Ashbaugh [2] (see also [3], [9]). Corollary 3.2.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…These inequalities are called universal because they do not involve domaindependencies [31] (see also [2], [3], for example).…”

Section: Introductionmentioning

confidence: 99%

“…Our work follows the spirit of [2] and [3]. In these works one of us (MSA) produced an argument partially based on the work of H.C. Yang [35], which does away with the "free parameters" of Hile-Protter [21] and Hook [23] (see also [31], [32]).…”

Section: Introductionmentioning

confidence: 99%

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A unified approach to universal inequalities for eigenvalues of elliptic operators

Ashbaugh

Hermi²

2004

Pacific J. Math.

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We present an abstract approach to universal inequalities for the discrete spectrum of a self-adjoint operator, based on commutator algebra, the Rayleigh-Ritz principle, and one set of "auxiliary" operators. The new proof unifies classical inequalities of Payne-Pólya-Weinberger, Hile-Protter, and H.C. Yang and provides a Yang type strengthening of Hook's bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the "free parameters" of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe.

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Section: Introductionmentioning

confidence: 54%

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…These inequalities are called universal because they do not involve domaindependencies [31] (see also [2], [3], for example).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A unified approach to universal inequalities for eigenvalues of elliptic operators

Ashbaugh

Hermi²

2004

Pacific J. Math.

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“…It is easy to prove that Yang's inequalities (1.5) and (1.6) are sharper than the inequality (1.4) of Hile and Protter (see [1] and [2] for details).…”

Section: Introductionmentioning

confidence: 99%

Inequalities for eigenvalues of a clamped plate problem

Cheng

Yang

2005

Trans. Amer. Math. Soc.

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Abstract. Let D be a connected bounded domain in an n-dimensional Euclidean space R n . Assume thatare eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator:Then, we give an upper bound of the (k +1)-th eigenvalue λ k+1 in terms of the first k eigenvalues, which is independent of the domain D, that is, we prove the following:Further, a more explicit inequality of eigenvalues is also obtained.

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Extrinsic eigenvalue estimates of Dirac operators on Riemannian manifolds

Huang¹,

Chen²,

Sun³

2011

Mathematische Nachrichten

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Key words Eigenvalue, generalized Dirac operator, isometric immersion MSC (2010) Primary: 35P15, Secondary: 58G25 For eigenvalues of generalized Dirac operators on compact Riemannian manifolds, we obtain a general inequality. By using this inequality, we study eigenvalues of generalized Dirac operators on compact submanifolds of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces. We obtain explicit bounds for the (k + 1)-th eigenvalue of generalized Dirac operators on such objects in terms of its first k eigenvalues, which depend on the mean curvature of the embedding and the curvature term in the Bochner-Weitzenböck formula for the square of the Dirac operator. These inequalities of eigenvalues extend the recent results in [15].

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The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H C Yang

Cited by 73 publications

References 20 publications

A unified approach to universal inequalities for eigenvalues of elliptic operators

A unified approach to universal inequalities for eigenvalues of elliptic operators

Inequalities for eigenvalues of a clamped plate problem

Extrinsic eigenvalue estimates of Dirac operators on Riemannian manifolds

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