Logarithmically convex function

Template:Short description In mathematics, a 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 composition of the logarithm with f, is itself a convex function.

Definition

Let Template:Math be a convex subset of a real vector space, and let Template:Math be a function taking non-negative values. Then Template:Math 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, Template:Math is logarithmically convex if and only if, for all Template:Math and all Template:Math, 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, Template:Math is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all Template:Math.

The above definition permits Template:Math to be zero, but if Template:Math is logarithmically convex and vanishes anywhere in Template:Math, then it vanishes everywhere in the interior of Template:Math.

Equivalent conditions

If Template:Math is a differentiable function defined on an interval Template:Math, then Template:Math is logarithmically convex if and only if the following condition holds for all Template:Math and Template:Math in Template:Math:

<math>\log f(x) \ge \log f(y) + \frac{f'(y)}{f(y)}(x - y).</math>

This is equivalent to the condition that, whenever Template:Math and Template:Math are in Template:Math and Template:Math,

<math>\left(\frac{f(x)}{f(y)}\right)^{\frac{1}{x - y}} \ge \exp\left(\frac{f'(y)}{f(y)}\right).</math>

Moreover, Template:Math is strictly logarithmically convex if and only if these inequalities are always strict.

If Template:Math is twice differentiable, then it is logarithmically convex if and only if, for all Template:Math in Template:Math,

<math>f(x)f(x) \ge f'(x)^2.</math>

If the inequality is always strict, then Template:Math is strictly logarithmically convex. However, the converse is false: It is possible that Template:Math is strictly logarithmically convex and that, for some Template:Math, we have <math>f(x)f(x) = f'(x)^2</math>. For example, if <math>f(x) = \exp(x^4)</math>, then Template:Math 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>Template:Harvnb.</ref><ref>Template:Harvnb.</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 composite of the 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

References

• John B. Conway. Functions of One Complex Variable I, second edition. Springer-Verlag, 1995. Template:Isbn.
• Template:Springer
• Template:Citation.

• Template:Citation.

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