Logarithmically convex function

In mathematics, a function f is logarithmically convex or superconvex if , the composition of the logarithm with f, is itself a convex function. 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. Here we interpret as . Explicitly, f is logarithmically convex if and only if, for all x1, x2 ∈ X and all t ∈ [0, 1], the two following equivalent conditions hold: 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. If f is a differentiable function defined on an interval I ⊆ R, then f is logarithmically convex if and only if the following condition holds for all x and y in I: This is equivalent to the condition that, whenever x and y are in I and x > y, Moreover, f is strictly logarithmically convex if and only if these inequalities are always strict. If f is twice differentiable, then it is logarithmically convex if and only if, for all x in I, If the inequality is always strict, then f is strictly logarithmically convex. However, the converse is false: It is possible that f is strictly logarithmically convex and that, for some x, we have . For example, if , then f is strictly logarithmically convex, but . Furthermore, is logarithmically convex if and only if is convex for all . If are logarithmically convex, and if are non-negative real numbers, then is logarithmically convex. If is any family of logarithmically convex functions, then is logarithmically convex. If is convex and is logarithmically convex and non-decreasing, then is logarithmically convex. A logarithmically convex function f is a convex function since it is the composite of the increasing convex function and the function , which is by definition convex.