Abstract
In this paper, we study the universal inequalities for eigenvalues of a clamped plate problem of the drifting Laplacian in several cases, and establish some universal inequalities that are different from those obtained previously in (Du et al. in Z Angew Math Phys 66(3):703–726, 2015).
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Yue He was partially supported by Key Laboratory of Applied Mathematics of Fujian Province University (Putian University) (No. SX202101).
Appendix
Appendix
To prove some results of this paper, we need the following inequalities:
Lemma 5.1
(Weighted Chebyshev inequality, see Hardy et al. 1988) Let \(\{a_i\}^k_{i=1},\{b_i\}^k_{i=1}\) and \(\{c_i\}^k_{i=1}\) be three sequences of non-negative real numbers with \(\{a_i\}^k_{i=1}\) decreasing; \(\{b_i\}^k_{i=1}\) and \(\{c_i\}^k_{i=1}\) increasing. Then the following inequality holds
Lemma 5.2
(Chebyshev sum inequality, see Hardy et al. 1988) Let \(\{a_i\}^k_{i=1}\) and \(\{b_i\}^k_{i=1}\) be two sequences of real numbers with \(\{a_i\}^k_{i=1}\) and \(\{b_i\}^k_{i=1}\) increasing or decreasing. Then the following inequality holds
with equality if and only if
Lemma 5.3
(Reverse Chebyshev inequality, see Hardy et al. 1988) Suppose \(\{a_i\}_{i=1}^k\) and \(\{b_i\}_{i=1}^k\) are two real sequences with \(\{a_i\}_{i=1}^k\) increasing and \(\{b_i\}_{i=1}^k\) decreasing. Then the following inequality holds:
with equality if and only if
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He, Y., Pu, S. Universal Inequalities for Eigenvalues of a Clamped Plate Problem of the Drifting Laplacian. Bull Braz Math Soc, New Series 55, 10 (2024). https://doi.org/10.1007/s00574-024-00384-w
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DOI: https://doi.org/10.1007/s00574-024-00384-w