Two elementary inequalities for real-valued polynomials

One generalization of (1), which is not quite an inequality, is the so-called Hawaiian conjecture. It states that the number of real zeroes of $(f'/f)'$ does not exceed the number of nonreal zeroes of $f$. This paper claims a proof: Mikhail Tyaglov, On the number of real critical points of logarithmic derivatives and the Hawaii conjecture, arXiv:0902.0413v3.

Regarding Newton's method for polynomials with all zeros real, there is some discussion in the paper by J. H. Hubbard, D. Schleicher and S. Sutherland:http://www.math.cornell.edu/~hubbard/NewtonInventiones.pdf, especially in Section 7. There is a reference to another paper by Barna there, and lots of other references, which may be useful.