On the betweenness condition of Rolle's theorem (Q1072666)

Let \(\Gamma =\{\gamma_ 0,\gamma_ 1,...\}\) be a sequence of real numbers. For any real polynomial \(f(x)=\sum a_ kx^ k,\) write \(\Gamma [f(x)]=\sum a_ k\gamma_ kx^ k.\) The sequence \(\Gamma\) is said to possess the betweenness property if the polynomial \(\Gamma [f(x)]\) has at least one real zero between any two real zeros of f(x). For the sequence \(\Gamma =\{0,1,2,...\},\) the betweenness property is a consequence of Rolle's theorem since \(\Gamma [f(x)]=xf'(x).\) The authors show that \(\Gamma\) has the betweenness property if and only if it is a nonconstant arithmetic sequence, all of whose terms have the same sign. Their main theorem is a quantitative result on the location of the zero c of \(\Gamma [f(x)]\) between two consecutive real zeros a and b of f, giving the best possible bounds for c depending only on \(\Gamma\),a,b and the degree of f. Their theorem generalizes a theorem of Laguerre for derivatives.