Abstract: I will firstly review the Szegö-Weinberger inequality and the Zhong-Yang inequality in Euclidean space. Secondly, I will present a Szegö–Weinberger inequality on curved manifolds and a Zhong-Yang inequality for the first non-zero eigenvalue of the weighted p-Laplacian on compact Bakry-Emery manifolds. Finally, I will discuss some open problems for the Neumann eigenvalues and eigenfunctions.
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