We derive upper bounds for the spectral radius of the $n\times n$ Hilbert matrix. The key idea is to write the Hilbert matrix as integral operator with positive kernel function and then to use a Wielandt-type min-max principle for the spectral radius. Choosing special trial functions yields a new bound which we show improves the best bound known heretofore.

Hilbert matrix, spectral radius, Wielandt min-max principle

15A42 (15A60, 47G10)