  We prove that for any circulant matrix $C$ of size $n\times n$ with the monic
characteristic polynomial $p(z)$, the spectrum of its $(n-1)\times(n-1)$
submatrix $C_{n-1}$ constructed with first $n-1$ rows and columns of $C$
consists of all critical points of $p(z)$. Using this fact we provide a simple
proof for the Schoenberg conjecture recently proved by R. Pereira and S.
Malamud. We also prove full generalization of a higher order Schoenberg-type
conjecture proposed by M. de Bruin and A. Sharma and recently proved by W.S.
Cheung and T.W. Ng. in its original form, i.e. for polynomials whose mass
centre of roots equals zero. In this particular case, our inequality is
stronger than it was conjectured by de Bruin and Sharma. Some Schmeisser's-like
results on majorization of critical point of polynomials are also obtained.
