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Proof of the Chen–Rubin conjecture

Published online by Cambridge University Press:  12 July 2007

Horst Alzer
Affiliation:
Morsbacher Straβe 10, 51545 Waldbröl, Germany(alzerhorst@freenet.de)

Abstract

Let n ≥ 0 be an integer and let λ(n) be the median of the Gamma distribution of order n + 1 with parameter 1. In 1986, Chen and Rubin conjectured that n ↦ λ (n) − n (n = 0, 1, 2, …) is decreasing. We prove the following monotonicity theorem, which settles this conjecture.

Let α and β be real numbers. The sequence n ↦ λ (n) – αn (n = 0, 1, 2, …) is strictly decreasing if and only if α; ≥ 1. And n ↦ λ(n) − βn (n = 0, 1, 2, …) is strictly increasing if and only if β < λ(1) − log 2 = 0.98519….

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2005

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