In this talk, we consider inequalities for the Riesz means of order γ for eigenvalues of Dirichlet and Neumann Laplacians. Classical results due to Berezin, Li–Yau, and Kröger state that for γ ≥ 1, the Riesz means are bounded by their semiclassical approximation. In recent joint work with R. Frank, we show that such inequalities extend to a range of γ <1, under the assumption that the domain is convex. I will present these extensions and discuss their implications in related shape optimization problems.
