Comparison Theorems for Eigenvalues of Elliptic Operators and the Generalized Polya Conjecture

We establish comparison theorems for eigenvalues between higher order elliptic equations on compact manifolds with boundary. As an application, it follows that if the Polya conjecture is true then so is the generalized Polya conjecture proposed by Ku et al. (J Differ Equ 97:127-139, 1992). We also obtain new lower bound for the eigenvalues of higher order elliptic equations on bounded domains in a Euclidean space.