Inmathematics, afunctionf islogarithmically convex orsuperconvex[1] if, thecomposition of thelogarithm withf, is itself aconvex function.
Definition
editLetX be aconvex subset of arealvector space, and letf :X →R be a function takingnon-negative values. Thenf 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 allx1,x2 ∈X and allt ∈ [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 allt ∈ (0, 1).
The above definition permitsf to be zero, but iff is logarithmically convex and vanishes anywhere inX, then it vanishes everywhere in the interior ofX.
Equivalent conditions
editIff is adifferentiable function defined on an intervalI ⊆R, thenf is logarithmically convex if and only if the following condition holds for allx andy inI:
This is equivalent to the condition that, wheneverx andy are inI andx >y,
Moreover,f is strictly logarithmically convex if and only if these inequalities are always strict.
Iff is twice differentiable, then it is logarithmically convex if and only if, for allx inI,
If the inequality is always strict, thenf is strictly logarithmically convex. However, the converse is false: It is possible thatf is strictly logarithmically convex and that, for somex, we have . For example, if , thenf is strictly logarithmically convex, but .
Furthermore, is logarithmically convex if and only if is convex for all .[2][3]
Sufficient conditions
editIf 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.
Properties
editA logarithmically convex functionf is a convex function since it is thecomposite of theincreasing convex function and the function , which is by definition convex. However, being logarithmically convex is a strictly stronger property than being convex. For example, the squaring function is convex, but its logarithm is not. Therefore the squaring function is not logarithmically convex.
Examples
edit- is logarithmically convex when and strictly logarithmically convex when .
- is strictly logarithmically convex on for all
- Euler'sgamma function is strictly logarithmically convex when restricted to the positive real numbers. In fact, by theBohr–Mollerup theorem, this property can be used to characterize Euler's gamma function among the possible extensions of thefactorial function to real arguments.
See also
editNotes
edit- ^Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.
- ^Montel 1928.
- ^NiculescuPersson 2006, p. 70.
References
edit- John B. Conway.Functions of One Complex Variable I, second edition. Springer-Verlag, 1995.ISBN 0-387-90328-3.
- "Convexity, logarithmic",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
- Niculescu, Constantin;Persson, Lars-Erik (2006),Convex Functions and their Applications - A Contemporary Approach (1st ed.),Springer,doi:10.1007/0-387-31077-0,ISBN 978-0-387-24300-9,ISSN 1613-5237.
- Montel, Paul (1928), "Sur les fonctions convexes et les fonctions sousharmoniques",Journal de Mathématiques Pures et Appliquées (in French),7:29–60.
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