Logarithmically convex function

In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {displaystyle {log }circ f} , the composition of the logarithm with f, is itself a convex function. In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {displaystyle {log }circ f} , 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: Here we interpret log ⁡ 0 {displaystyle log 0} as − ∞ {displaystyle -infty } . Explicitly, f is logarithmically convex if and only if, for all x1, x2 ∈ X and all t ∈ , 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.