Ann Oper Res https://doi.org/10.1007/s10479-018-2923-y S.I.: STOCHASTIC MODELING AND OPTIMIZATION, IN MEMORY OF ANDRÁS PRÉKOPA Logconcavity, twice-logconcavity and Turán-type inequalities 1 2 Anh Ninh · Minh Pham © Springer Science+Business Media, LLC, part of Springer Nature 2018 Abstract This paper presents a brief review of the logconcavity of some compound distri- butions and its relationship to Turán-type inequalities. Further, we introduce a new concept of twice-logconcavity (twice-logconvexity) and show that many discrete probability distri- butions and combinatorial sequences are twice-logconcave. These results are closely related to other areas of mathematics such as combinatorics and analysis. Keywords Compound Poisson · Logconcavity · Turán inequalities · Divided differences 1 Introduction In 1912, Fekete introduced the notion of an r-times positive sequence, which is known as twice-positive sequence, or, log-concave sequence for r = 2. A sequence of nonnegative elements... a , a , a ,... is said to be r-times positive if the matrix −2 −1 0 ⎛ ⎞ . . . . . . . . . ⎜ ⎟ ⎜ ⎟ ⎜ . ⎟ a a a ⎜ 0 1 2 ⎟ ⎜ ⎟ . . ⎜ ⎟ A = . . , . . ⎜ a a a ⎟ −1
Annals of Operations Research – Springer Journals
Published: Jun 6, 2018
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