Function whose composition with the logarithm is convex
In mathematics, a function "f" is logarithmically convex or superconvex if formula_1, the composition of the logarithm with "f", is itself a convex function.
Definition.
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 formula_4 as formula_5.
Explicitly, "f" is logarithmically convex if and only if, for all "x"1, "x"2 ∈ "X" and all "t" ∈ [0, 1], the two following equivalent conditions hold:
formula_6
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".
Equivalent conditions.
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":
formula_7
This is equivalent to the condition that, whenever "x" and "y" are in "I" and "x" ; "y",
formula_8
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",
formula_9
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 formula_10. For example, if formula_11, then "f" is strictly logarithmically convex, but formula_12.
Furthermore, formula_13 is logarithmically convex if and only if formula_14 is convex for all formula_15.
Sufficient conditions.
If formula_16 are logarithmically convex, and if formula_17 are non-negative real numbers, then formula_18 is logarithmically convex.
If formula_19 is any family of logarithmically convex functions, then formula_20 is logarithmically convex.
If formula_21 is convex and formula_22 is logarithmically convex and non-decreasing, then formula_23 is logarithmically convex.
Properties.
A logarithmically convex function "f" is a convex function since it is the composite of the increasing convex function formula_24 and the function formula_25, which is by definition convex. However, being logarithmically convex is a strictly stronger property than being convex. For example, the squaring function formula_26 is convex, but its logarithm formula_27 is not. Therefore the squaring function is not logarithmically convex.
Notes.
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References.
"This article incorporates material from logarithmically convex function on PlanetMath, which is licensed under the ."