Proof of the Chen–Rubin conjecture

Author: Alzer, Horst

Source: Proceedings Section A: Mathematics - Royal Society of Edinburgh, Volume 135, Number 4, August 2005 , pp. 677-688(12)

Publisher: Royal Society of Edinburgh

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Abstract:

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

Let alpha and beta be real numbers. The sequence n map lambda (n) – alphan (n = 0, 1, 2, …) is strictly decreasing if and only if alpha ges 1. And n map lambda(n) - betan (n = 0, 1, 2, …) is strictly increasing if and only if beta < lambda(1) - log 2 = 0.98519….

Document Type: Research article

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