View source for Logarithmically convex function

{{Short description|Function whose composition with the logarithm is convex}}
In [[mathematics]], a [[function (mathematics)|function]] ''f'' is '''logarithmically convex''' or '''superconvex'''<ref>Kingman, J.F.C.  1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.</ref> if <math>{\log}\circ f</math>, the [[function composition|composition]] of the [[logarithm]] with ''f'', is itself a [[convex function]].

==Definition==
Let {{math|''X''}} be a [[convex set|convex subset]] of a [[real numbers|real]] [[vector space]], and let {{math|''f'' : ''X'' → '''R'''}} be a function taking [[negative and positive numbers|non-negative]] values.  Then {{math|''f''}} is:
* '''Logarithmically convex''' if <math>{\log} \circ f</math> is convex, and
* '''Strictly logarithmically convex''' if <math>{\log} \circ f</math> is strictly convex.
Here we interpret <math>\log 0</math> as <math>-\infty</math>.

Explicitly, {{math|''f''}} is logarithmically convex if and only if, for all {{math|''x''<sub>1</sub>, ''x''<sub>2</sub> ∈ ''X''}} and all {{math|''t'' ∈ [0, 1]}}, the two following equivalent conditions hold:
:<math>\begin{align}
\log f(tx_1 + (1 - t)x_2) &\le t\log f(x_1) + (1 - t)\log f(x_2), \\
f(tx_1 + (1 - t)x_2) &\le f(x_1)^tf(x_2)^{1-t}.
\end{align}</math>
Similarly, {{math|''f''}} is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all {{math|''t'' ∈ (0, 1)}}.

The above definition permits {{math|''f''}} to be zero, but if {{math|''f''}} is logarithmically convex and vanishes anywhere in {{math|''X''}}, then it vanishes everywhere in the interior of {{math|''X''}}.

===Equivalent conditions===
If {{math|''f''}} is a differentiable function defined on an interval {{math|''I'' ⊆ '''R'''}}, then {{math|''f''}} is logarithmically convex if and only if the following condition holds for all {{math|''x''}} and {{math|''y''}} in {{math|''I''}}:
:<math>\log f(x) \ge \log f(y) + \frac{f'(y)}{f(y)}(x - y).</math>
This is equivalent to the condition that, whenever {{math|''x''}} and {{math|''y''}} are in {{math|''I''}} and {{math|''x'' > ''y''}},
:<math>\left(\frac{f(x)}{f(y)}\right)^{\frac{1}{x - y}} \ge \exp\left(\frac{f'(y)}{f(y)}\right).</math>
Moreover, {{math|''f''}} is strictly logarithmically convex if and only if these inequalities are always strict.

If {{math|''f''}} is twice differentiable, then it is logarithmically convex if and only if, for all {{math|''x''}} in {{math|''I''}},
:<math>f''(x)f(x) \ge f'(x)^2.</math>
If the inequality is always strict, then {{math|''f''}} is strictly logarithmically convex.  However, the converse is false: It is possible that {{math|''f''}} is strictly logarithmically convex and that, for some {{math|''x''}}, we have <math>f''(x)f(x) = f'(x)^2</math>.  For example, if <math>f(x) = \exp(x^4)</math>, then {{math|''f''}} is strictly logarithmically convex, but <math>f''(0)f(0) = 0 = f'(0)^2</math>.

Furthermore, <math>f\colon I \to (0, \infty)</math> is logarithmically convex if and only if <math>e^{\alpha x}f(x)</math> is convex for all <math>\alpha\in\mathbb R</math>.<ref>{{harvnb|Montel|1928}}.</ref><ref>{{harvnb|NiculescuPersson|2006|p=70}}.</ref>

==Sufficient conditions==
If <math>f_1, \ldots, f_n</math> are logarithmically convex, and if <math>w_1, \ldots, w_n</math> are non-negative real numbers, then <math>f_1^{w_1} \cdots f_n^{w_n}</math> is logarithmically convex.

If <math>\{f_i\}_{i \in I}</math> is any family of logarithmically convex functions, then <math>g = \sup_{i \in I} f_i</math> is logarithmically convex.

If <math>f \colon X \to I \subseteq \mathbf{R}</math> is convex and <math>g \colon I \to \mathbf{R}_{\ge 0}</math> is logarithmically convex and non-decreasing, then <math>g \circ f</math> is logarithmically convex.

==Properties==
A logarithmically convex function ''f'' is a convex function since it is the [[function composition|composite]] of the [[increasing function|increasing]] convex function <math>\exp</math> and the function <math>\log\circ f</math>, which is by definition convex.  However, being logarithmically convex is a strictly stronger property than being convex.  For example, the squaring function <math>f(x) = x^2</math> is convex, but its logarithm <math>\log f(x) = 2\log |x|</math> is not.  Therefore the squaring function is not logarithmically convex.

==Examples==
* <math>f(x) = \exp(|x|^p)</math> is logarithmically convex when <math>p \ge 1</math> and strictly logarithmically convex when <math>p > 1</math>.
* <math>f(x) = \frac{1}{x^p}</math> is strictly logarithmically convex on <math>(0,\infty)</math> for all <math>p>0.</math>
* Euler's [[gamma function]] is strictly logarithmically convex when restricted to the positive real numbers.  In fact, by the [[Bohr–Mollerup theorem]], this property can be used to characterize Euler's gamma function among the possible extensions of the [[factorial]] function to real arguments.

==See also==
* [[Logarithmically concave function]]

==Notes==
{{Reflist}}

==References==
* John B. Conway. ''Functions of One Complex Variable I'', second edition. Springer-Verlag, 1995. {{isbn|0-387-90328-3}}.
* {{springer|title=Convexity, logarithmic|id=p/c026410}}
* {{citation
 | last1 = Niculescu
 | first1 = Constantin
 | last2 = Persson
 | first2 = Lars-Erik
 | author2-link = Lars-Erik Persson
 | title = Convex Functions and their Applications - A Contemporary Approach
 | publisher = [[Springer-Verlag|Springer]]
 | year = 2006
 | edition = 1st
 | language = English
 | doi = 10.1007/0-387-31077-0
 | isbn = 978-0-387-24300-9
 | issn = 1613-5237
}}.

* {{citation
 | last1 = Montel
 | first1 = Paul
 | author1-link = Paul Montel
 | title = Sur les fonctions convexes et les fonctions sousharmoniques
 | journal = Journal de Mathématiques Pures et Appliquées
 | year = 1928
 | language = French
 | pages = 29–60
 | volume = 7
}}.

{{Convex analysis and variational analysis}}

{{PlanetMath attribution|id=5664|title=logarithmically convex function}}

[[Category:Real analysis]]