Location of zeros. I: Real polynomials and entire functions
This paper represents the first portion of a two-part investigation concerned with distribution of zeros of polynomials and entire functions. Let \(\Gamma =\{\gamma_ k\}^{\infty}_{k=0}\) be a real sequence. For an arbitrary real polynomial \(f(x)=\sum^{n}_{k=0} a_ k x^ k\), we define \(\Gamma(f)\) by \(\Gamma(f(x)) = \sum^{n}_{k=0} a_ k \gamma_ k x^ k\). A sequence \(\Gamma\) is said to be a multiplier sequence (of the first kind) if it takes every polynomial f(x) with only real zeros into a polynomial \(\Gamma\) (f(x)) of the same class. In section 2 the authors present a new proof of the fundamental inequality which states that the real sequence \(\Gamma\) is a multiplier sequence if and only if \(Z_ c(\Gamma(f))\leq Z_ c(f)\) for all real polynomials f, where \(Z_ c(f)\) denotes the number of non-real zeros of the polynomial f, counting multiplicities. A real sequence \(\Gamma =\{\gamma_ 0,...,\gamma_ n\}\) is called an n- sequence if for every polynomial f(x) of degree less than or equal to n and with only real zeros the polynomial \(\Gamma\) (f(x)) also has only real zeros. In section 3 the authors give several characterizations of n- sequences. One of them states: \(\Gamma\) is an n-sequence if and only if the zeros of the polynomial \(\Gamma [(1+x)^ n]\) are all real and of the same sign. In this section the following interesting fact is also proved: If \(\phi(x)=\sum^{n}_{k=0}\gamma_ k x^ k/k!,\quad \gamma_ k\geq 0,\quad n\geq 2\) and all zeros of \(\phi\) lie in the set \(S=\{z=x+iy:| y| \leq | x| /\sqrt{n-1}\}\) then \(\Gamma =\{\gamma_ 0,...,\gamma_ n\}\) is an n-sequence. In section 4 the extendability of n-sequences is examined. In particular, it is shown that for every \(n\geq 3\) there is an n-sequence which does not extend to an \((n+1)\)-sequence, and every 2-sequence is extendable to a multiplier sequence.
location of zeros of real polynomials and entire functions
scientific article; zbMATH DE number 3785226