Universal Inequalities for Eigenvalues of a Clamped Plate Problem of the Drifting Laplacian

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).

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

$$\begin{aligned} \left( \sum _{i=1}^ka_i^2b_i\right) \left( \sum _{i=1}^ka_ic_i\right) \le \left( \sum _{i=1}^ka_i^2\right) \left( \sum _{i=1}^ka_ib_ic_i\right) . \end{aligned}$$

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

$$\begin{aligned} \frac{1}{k}\left( \sum _{i=1}^ka_i\right) \left( \sum _{i=1}^kb_i\right) \le \sum _{i=1}^ka_ib_i. \end{aligned}$$

with equality if and only if

$$\begin{aligned} a_1=\cdots =a_k,\quad \hbox {or}\quad b_1=\cdots =b_k. \end{aligned}$$

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:

$$\begin{aligned} \frac{1}{k}\left( \sum _{i=1}^ka_i\right) \left( \sum _{i=1}^kb_i\right) \ge \sum _{i=1}^ka_ib_i, \end{aligned}$$

with equality if and only if

$$\begin{aligned} a_1=\cdots =a_k\quad \hbox {or}\quad b_1=\cdots =b_k. \end{aligned}$$