On sequences that do not increase the number of real roots of polynomials

A complete description is given for the sequences {~'k}~=0 such that, for an arbitrary real polynomial f ( t) = k=oakt, an arbitrary A ~ (0,+~), anda fixed C ~ (0, +~), the number of roots of the polynomial (Tf) (t) = ~2=0 ak)~k tk on [0, C] does not exceed the number of roots of f(t) on [0, A]. The following problem was formulated in [ 1, p. 382]: Karlin's problem. Describe the sequences of factors {)~k}k = 0 that do not increase the number of real zeros of n polynomials, i.e., the sequences such that, for any real polynomial f (x) = ~,k=0 akxk' k=0 k=0 (1) where ~lg ( f) is the number of real zeros of f taking account of their multiplicities. In [6], it was proved that the solution of this problem presented in [2-5] is not correct and, thus, Karlin's prob-lem remains open. In this paper, we describe the sequences of factors {)~k}k = 0 such that, for an arbitrary real polynomial f ( t) = ~nk=o akt k, an arbitrary A ~ (0, oo), and a fixed C e (0, ~,), the number of roots of the polynomial ~=0 ak)~kx~ on [0, C] does not exceed the number of roots of f ( t) on [0, A]. Denote by " ~ the class of all sequences that do not increase the number of real zeros, i.e., sequences satisfying n property (1); the transformations determined by sequences of this type are denoted by T, i.e., if f (x) = ~=0 akx~' then Let us prove some auxiliary results. n (T f) (x) = ~. ~ ak)~kX k. (2) k=0 Lemma 1. If{X.k}~__o e % = + _ 1 and ~ = +. 1 such that then there exists a nondecreasing function g(t) on o