A necessary and sufficient condition that a vector f is an antieigenvector of a strictly accretive operator A is obtained. The structure of antieigenvectors of selfadjoint and certain class of normal operators is also found in terms of eigenvectors. The Kantorovich inequality for selfadjoint operators and the Davis's inequality for normal operators are then easily deduced. A sort of uniqueness is also established for the values of Re ( A f , f ) and ‖ A f ‖ if the first antieigenvalue, which is equal to min Re ( A f , f ) / ( ‖ A f ‖ ‖ f ‖ ) is attained at the unit vector f .