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Let X be a convex subset of a real vector space, and let f : X → R be a function taking non-negative values. Then f is: . Logarithmically convex if is convex, and; Strictly logarithmically convex if is strictly convex.
In convex analysis, a non-negative function f : R n → R + is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality (+ ()) () for all x,y ∈ dom f and 0 < θ < 1.
A graph of the bivariate convex function x 2 + xy + y 2. Convex vs. Not convex. In mathematics, ... is logarithmically convex if is a convex function. The term ...
The rows of Pascal's triangle are examples for logarithmically concave sequences. In mathematics, a sequence a = (a 0, a 1, ..., a n) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if a i 2 ≥ a i−1 a i+1 holds for 0 < i < n.
When a some complete Reinhardt domain to be the domain of convergence of a power series, an additional condition is required, which is called logarithmically-convex. A Reinhardt domain D is called logarithmically convex if the image () of the set
Thus, the collection of −∞-convex measures is the largest such class, whereas the 0-convex measures (the logarithmically concave measures) are the smallest class. The convexity of a measure μ on n-dimensional Euclidean space R n in the sense above is closely related to the convexity of its probability density function. [2]
There are three versions of the conjecture, one in terms of logarithmically convex functions, one in terms of increasing functions, and one in terms of non-negative functions. The conjecture has implications in the study of complex functions and is related to Euler's Beta function. While the conjecture is known to hold for certain conditions ...
There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive (Bohr–Mollerup theorem).