Numerical radius and zero pattern of matrices

Let A   be an n×nn×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n  -vector. Then, letting ‖A‖‖A‖ be the Frobenius norm of A, we show that|〈Ax,x〉|2⩽(1−1/2r−1/2n)‖A‖2.|〈Ax,x〉|2⩽(1−1/2r−1/2n)‖A‖2. This inequality is tight within an additive term O(rn−2)O(rn−2). If the matrix A is Hermitian, then|〈Ax,x〉|2⩽(1−1/r)‖A‖2.|〈Ax,x〉|2⩽(1−1/r)‖A‖2. This inequality is sharp; moreover, it implies the Turán theorem for graphs.