Separation of zeros of translates of polynomials and entire functions. (English) Zbl 0880.30008
For an entire function \(f\) with zeros (a), \(\delta(f)= \inf\{|a_i-a_j|: i\neq j\}\) is called the separation of zeros. In his earlier paper [Am. Math. Mon. 100, No. 3, 272-273 (1993; Zbl 0767.26010)] the author showed that \(\delta (f-kf) >\delta (f)\) for all \(k \mathbb{R}\), and in two later papers [J. Aust. Math. Soc., Ser. A 59, No. 2, 330-342 (1995; Zbl 0873.26006), Proc. Edinb. Math. Soc., II. Ser. 39, No. 3, 357-363 (1996; Zbl 0853.26009)] gave estimates of the increase of the separation of zeros. In the present paper the author gives the extensions of these results to entire functions of order less than 2.
Reviewer: J.Matkowski (Bielsko-Biała)
MSC:
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 26C10 | Real polynomials: location of zeros |
References:
| [1] | Gelca, R., A short proof of a result on polynomials, Amer. Math. Monthly, 100, 936-937 (1993) · Zbl 0875.26027 |
| [2] | Pólya, G., Bemerkung über die Integraldarstellung der Riemannsche ξ-Funktion, Acta Math., 48, 305-317 (1926) · JFM 52.0335.02 |
| [3] | Titchmarsh, E. C., The Theory of Functions (1939), Oxford University Press: Oxford University Press Oxford, p. 266-266 · Zbl 0022.14602 |
| [4] | Walker, P. L., Separation of the zeros of polynomials, Amer. Math. Monthly, 100, 272-273 (1993) · Zbl 0767.26010 |
| [5] | Walker, P. L., Bounds for the separation of real zeros of polynomials, J. Austral. Math. Soc. (Ser. A), 59, 330-342 (1995) · Zbl 0873.26006 |
| [6] | Walker, P. L., Upper bounds for the separation of the zeros of polynomials, Proc. Edinburgh Math. Soc., 39, 357-363 (1996) · Zbl 0853.26009 |
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