The authors study the fourth order linear operator
coupled with the clamped beam conditions
. They obtain the exact values of the real parameter
for which this operator satisfies an anti-maximum principle. When
, they obtain the best estimate by means of the spectral theory and, for
, they attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation
. By using the method of lower and upper solutions, they also prove the existence of solutions of nonlinear problems coupled with these boundary conditions.