A transform of finite order entire functions and perturbations of zeros

Abstract

A new transform of finite order entire functions is introduced. In terms of that transform and the Taylor coefficients, perturbation results are established. In addition, the conditions that provide conservation of the total multiplicities of the zeros in some domains are investigated. The results obtained in this article enable us to estimate the distances between the zeros of entire functions and the zeros of their derivatives. They also enable us to explore the relations between the zeros of power series, their partial sums and tails.

Keywords:

• finite order entire functions
• perturbations, partial sums
• tails of the Taylor series
• bounds for zeros
• critical points
• conservation of multiplicities

AMS Subject Classifications:

• 30C15
• 30D10

Acknowledgement

This research was supported by the Kamea Fund of Israel.