## Abstract

Let f be an entire function. Denote by z_{1}(f),z _{2}(f),... the zeros of f with their multiplicities. In the paper, estimates for the sums σ_{k=1}^{j1} _{|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 S_{p}(f):=σ_{k=1}^{∞} 1/zk(f)|p (p≥;p(f)) and σ_{k=1}^{∞} (Im 1/zk(f)) ^{2} (p(f)<2), as well as relations between the series σ_{k=1}^{∞1}z_{k}^{m}(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 language | English |
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Pages (from-to) | 117-159 |

Number of pages | 43 |

Journal | Acta Applicandae Mathematicae |

Volume | 99 |

Issue number | 2 |

DOIs | |

State | Published - 1 Nov 2007 |

## Keywords

- Bounds for zeros
- Counting function
- Entire functions