On eigenvalues and boundary curvature of the numerical range

Let A be an n x n matrix. By Donoghue's theorem, all corner points of its n umerical range W(A) belong to the spectrum sigma (A). It is therefore natur al to expect that, more generally, the distance from a point rho on the bou ndary partial derivativeW(A) of W(A) to partial derivative (A) should be in some sense bounded by the radius of curvature of partial derivativeW(A) at rho. We establish some quantitative results in this direction. (C) 2001 El sevier Science Inc. All rights reserved.