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Logarithmically convex function

Inmathematics, afunctionf islogarithmically convex orsuperconvex[1] iflogf{\displaystyle {\log }\circ f}, thecomposition of thelogarithm withf, is itself aconvex function.

Definition

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LetX be aconvex subset of arealvector space, and letf :XR be a function takingnon-negative values. Thenf is:

Here we interpretlog0{\displaystyle \log 0}  as{\displaystyle -\infty } .

Explicitly,f is logarithmically convex if and only if, for allx1,x2X and allt ∈ [0, 1], the two following equivalent conditions hold:

logf(tx1+(1t)x2)tlogf(x1)+(1t)logf(x2),f(tx1+(1t)x2)f(x1)tf(x2)1t.{\displaystyle {\begin{aligned}\log f(tx_{1}+(1-t)x_{2})&\leq t\log f(x_{1})+(1-t)\log f(x_{2}),\\f(tx_{1}+(1-t)x_{2})&\leq f(x_{1})^{t}f(x_{2})^{1-t}.\end{aligned}}} 

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

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Iff is adifferentiable function defined on an intervalIR, thenf is logarithmically convex if and only if the following condition holds for allx andy inI:

logf(x)logf(y)+f(y)f(y)(xy).{\displaystyle \log f(x)\geq \log f(y)+{\frac {f'(y)}{f(y)}}(x-y).} 

This is equivalent to the condition that, wheneverx andy are inI andx >y,

(f(x)f(y))1xyexp(f(y)f(y)).{\displaystyle \left({\frac {f(x)}{f(y)}}\right)^{\frac {1}{x-y}}\geq \exp \left({\frac {f'(y)}{f(y)}}\right).} 

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,

f(x)f(x)f(x)2.{\displaystyle f''(x)f(x)\geq f'(x)^{2}.} 

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 havef(x)f(x)=f(x)2{\displaystyle f''(x)f(x)=f'(x)^{2}} . For example, iff(x)=exp(x4){\displaystyle f(x)=\exp(x^{4})} , thenf is strictly logarithmically convex, butf(0)f(0)=0=f(0)2{\displaystyle f''(0)f(0)=0=f'(0)^{2}} .

Furthermore,f:I(0,){\displaystyle f\colon I\to (0,\infty )}  is logarithmically convex if and only ifeαxf(x){\displaystyle e^{\alpha x}f(x)}  is convex for allαR{\displaystyle \alpha \in \mathbb {R} } .[2][3]

Sufficient conditions

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Iff1,,fn{\displaystyle f_{1},\ldots ,f_{n}}  are logarithmically convex, and ifw1,,wn{\displaystyle w_{1},\ldots ,w_{n}}  are non-negative real numbers, thenf1w1fnwn{\displaystyle f_{1}^{w_{1}}\cdots f_{n}^{w_{n}}}  is logarithmically convex.

If{fi}iI{\displaystyle \{f_{i}\}_{i\in I}}  is any family of logarithmically convex functions, theng=supiIfi{\displaystyle g=\sup _{i\in I}f_{i}}  is logarithmically convex.

Iff:XIR{\displaystyle f\colon X\to I\subseteq \mathbf {R} }  is convex andg:IR0{\displaystyle g\colon I\to \mathbf {R} _{\geq 0}}  is logarithmically convex and non-decreasing, thengf{\displaystyle g\circ f}  is logarithmically convex.

Properties

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A logarithmically convex functionf is a convex function since it is thecomposite of theincreasing convex functionexp{\displaystyle \exp }  and the functionlogf{\displaystyle \log \circ f} , which is by definition convex. However, being logarithmically convex is a strictly stronger property than being convex. For example, the squaring functionf(x)=x2{\displaystyle f(x)=x^{2}}  is convex, but its logarithmlogf(x)=2log|x|{\displaystyle \log f(x)=2\log |x|}  is not. Therefore the squaring function is not logarithmically convex.

Examples

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See also

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Notes

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  1. ^Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.
  2. ^Montel 1928.
  3. ^NiculescuPersson 2006, p. 70.

References

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  • Montel, Paul (1928), "Sur les fonctions convexes et les fonctions sousharmoniques",Journal de Mathématiques Pures et Appliquées (in French),7:29–60.

This article incorporates material from logarithmically convex function onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.


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