Starting from a doubly infinite sequence of complex numbers, the aim of this paper is to extend certain Markov inequalities for the determinant of Hankel matrices and the zeros of the corresponding orthogonal polynomials on the real line (A. Markov in Notes of the Imperial Academy of Sciences, St. Petersburg, 74 (Appendix n. 2) (1894) 1-30. English translation, by J. Shohat, Duke Math. J. 7 (1940), 85-96) to the Toeplitz case, where the central role is played by CD kernels and paraorthogonal polynomials on the unit circle. In particular, we consider the case in which the starting sequence is a two-sided Pólya frequency sequence.