Some Results for Discrete Unimodality

By J. Keilson, Hans U. Gerber. 1971

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Log-Concavity and Strong Log-Concavity: a review (2014, background): “In connection with part (a) of this Theorem, Keilson and Gerber (1971) proved more: analogously to Ibragimov (1956a) they showed that p is strongly unimodal (i.e.”
The independence polynomial of a graph - a survey (2005, background): “7 ([53]) If P (x) is log-concave and Q(x) is unimodal, then P (x) · Q(x) is unimodal, while the product of two log-concave polynomials is log-concave.”
A new discrete distribution (2014, methodology): “Therefore, the result follows using Theorem 3 of [40].”
A new discrete probability distribution with integer support on (−∞, ∞) (2016, background): “As a direct consequence of log-concavity (Mark, 1996, Proposition 10, also see Kelison and Gerber, 1971; Gupta et al., 1997), the proposed DLOG (p, μ) distribution: i. has an increasing failure (hazard) rate distribution; ii. is strongly unimodal; iii. has all its moments; iv. remains log-concave…”
A Discrete Analog of the Generalized Exponential Distribution (2012, background): “According to Keilson and Gerber (1971), it is sufficient to show that for ≥ 1, p2 y ≥ py−1py+1 and for ≤ 1, p2 y ≤ py−1py+1 for all y = 2 3 .”