Strengthening Kröger’s inequality by including a correction term

Strengthen Kröger’s upper bounds for Neumann Laplacian eigenvalues on bounded domains by deriving an inequality that includes a quantitative correction term with explicit constants, thereby improving the original Kröger inequality for sums or individual Neumann eigenvalues.

Background

The authors cite an open question posed by Weidl in 2006 asking whether Kröger’s inequality can be strengthened by adding a correction term. They note that previous works provided improvements but without explicit constants. The present paper claims to resolve this question by presenting an improved Kröger inequality with an explicit correction term and constants.

While this question is historically open as stated, the authors’ main theorem in Section 3 supplies an explicit strengthened bound, indicating the question is resolved within the scope of this paper.