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

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

• Logarithmically convex if${\displaystyle \log \circ f}$ is convex, and
• Strictly logarithmically convex if${\displaystyle \log \circ f}$ is strictly convex.

Here we interpret${\displaystyle \log 0}$ as${\displaystyle -\mathrm{\infty }}$.

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.

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

${\displaystyle \log f(x)\ge \log f(y)+\frac{f^{\prime }(y)}{f(y)}(x-y).}$

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

${\displaystyle {\left(\frac{f(x)}{f(y)}\right)}^{\frac{1}{x-y}}\ge \exp \left(\frac{f^{\prime }(y)}{f(y)}\right).}$

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,

${\displaystyle f^{″}(x)f(x)\ge f^{\prime }(x{)}^2.}$

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 have${\displaystyle f^{″}(x)f(x)=f^{\prime }(x{)}^2}$. For example, if${\displaystyle f(x)=\exp (x^4)}$, thenf is strictly logarithmically convex, but${\displaystyle f^{″}(0)f(0)=0=f^{\prime }(0{)}^2}$.

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

If${\displaystyle f_1,\dots ,f_n}$ are logarithmically convex, and if${\displaystyle w_1,\dots ,w_n}$ are non-negative real numbers, then${\displaystyle f_1^{w_1}\cdots f_n^{w_n}}$ is logarithmically convex.

If${\displaystyle \{f_i{\}}_{i\in I}}$ is any family of logarithmically convex functions, then${\displaystyle g=\underset{i\in I}{sup}{f}_i}$ is logarithmically convex.

If${\displaystyle f:X\to I\subseteq \mathbf{R}}$ is convex and${\displaystyle g:I\to {\mathbf{R}}_{\ge 0}}$ is logarithmically convex and non-decreasing, then${\displaystyle g\circ f}$ is logarithmically convex.

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

• ${\displaystyle f(x)=\exp (|x{|}^p)}$ is logarithmically convex when${\displaystyle p\ge 1}$ and strictly logarithmically convex when${\displaystyle p>1}$.
• ${\displaystyle f(x)=\frac{1}{x^p}}$ is strictly logarithmically convex on${\displaystyle (0,\mathrm{\infty })}$ for all${\displaystyle p>0.}$
• 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.

• Logarithmically concave function

• 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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