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Similarly, f is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all t ∈ (0, 1). The above definition permits f to be zero, but if f is logarithmically convex and vanishes anywhere in X, then it vanishes everywhere in the interior of X.
for all x,y ∈ dom f and 0 < θ < 1. Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is log-convex if it satisfies the reverse inequality
A graph of the bivariate convex function x 2 + xy + y 2. Convex vs. Not convex. In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph between the two points.
Since −log(x) is a strictly convex function for x > 0, it follows that equality holds when p(x) equals q(x) almost everywhere. Rao–Blackwell theorem
The Bohr–Mollerup theorem states that among all functions extending the factorial functions to the positive real numbers, only the gamma function is log-convex, that is, its natural logarithm is convex on the positive real axis. Another characterisation is given by the Wielandt theorem.
Because of our relation for Γ(x + n), if we can fully understand Γ(x) for 0 < x ≤ 1 then we understand Γ(x) for all values of x. For x 1, x 2, the slope S(x 1, x 2) of the line segment connecting the points (x 1, log(Γ (x 1))) and (x 2, log(Γ (x 2))) is monotonically increasing in each argument with x 1 < x 2 since we have stipulated ...
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LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function [7] by adding an extra argument set to zero: