Note on the local growth of iterated polynomials
Let f be a real polynomial and let \(x_ i\), for \(i=1,2,3,...\), be defined by \(x_{i+1}=f(x_ i),\) where \(x_ i\) is a real number. The authors have proved that if f is of degree n and \(x_{i+1}-x_ i\geq 1\) for \(i=1,...,n+1\) then, for every \(i\in \{1,...,n\}\) and \(k\in \{1,...,n- i+1\},\) \[ -2^{k-1}/k!<f[x_ i,...,x_{i+k}]<(x_{i+k+1}- x_{i+k}+2^{k-1}-1)/k!, \] where \(f[x_ i,...,x_{i+k}]\) denotes the Newton difference quotient for f and \(x_ i,...,x_{i+k}.\)- As a consequence of this theorem the authors have obtained certain bounds on the local increase or decrease of the iteration sequence \((x_ i: i=1,2,3,...).\) Also a number-theoretic result related to a generalization of the van der Waerden numbers has been given as another consequence of the quoted theorem.
scientific article; zbMATH DE number 3880969