Bounds for zeros of entire functions

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Abstract

Let f be an entire function. Denote by z1(f),z 2(f),... the zeros of f with their multiplicities. In the paper, estimates for the sums σk=1j1 |zk(f)|(j=1,2,...) and for the counting function of the zeros of f are established. If f is of finite order ρ(f), we derive bounds for the series Sp(f):=σk=1 1/zk(f)|p (p≥;p(f)) and σk=1 (Im 1/zk(f)) 2 (p(f)<2), as well as relations between the series σk=1∞1zkm(f) (m≥;p(f)) and the traces of certain matrices. The contents of the paper is closely connected with the following well-known results: the Hadamard theorem on the convergence exponent of the zeros of an entire function, the Jensen inequality for the counting function, the Cauchy theorem on the comparison of the zeros of polynomials, Ostrowski's inequalities for the real and imaginary parts of the zeros of polynomials and the Cartwright-Levinson theorem. The suggested approach is based on the recent development of the spectral theory of linear operators.

Original languageEnglish
Pages (from-to)117-159
Number of pages43
JournalActa Applicandae Mathematicae
Volume99
Issue number2
DOIs
StatePublished - 1 Nov 2007

Keywords

  • Bounds for zeros
  • Counting function
  • Entire functions

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