Logarithmic Convexity
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a function ''f'' is logarithmically convex or superconvex if \circ f, the composition of the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
with ''f'', is itself a
convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigra ...
.


Definition

Let be a
convex subset In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
of a real
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, and let be a function taking non-negative values. Then is: * Logarithmically convex if \circ f is convex, and * Strictly logarithmically convex if \circ f is strictly convex. Here we interpret \log 0 as -\infty. Explicitly, is logarithmically convex if and only if, for all and all , the two following equivalent conditions hold: :\begin \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)^. \end Similarly, is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all . The above definition permits to be zero, but if is logarithmically convex and vanishes anywhere in , then it vanishes everywhere in the interior of .


Equivalent conditions

If is a differentiable function defined on an interval , then is logarithmically convex if and only if the following condition holds for all and in : :\log f(x) \ge \log f(y) + \frac(x - y). This is equivalent to the condition that, whenever and are in and , :\left(\frac\right)^ \ge \exp\left(\frac\right). Moreover, is strictly logarithmically convex if and only if these inequalities are always strict. If is twice differentiable, then it is logarithmically convex if and only if, for all in , :f''(x)f(x) \ge f'(x)^2. If the inequality is always strict, then is strictly logarithmically convex. However, the converse is false: It is possible that is strictly logarithmically convex and that, for some , we have f''(x)f(x) = f'(x)^2. For example, if f(x) = \exp(x^4), then is strictly logarithmically convex, but f''(0)f(0) = 0 = f'(0)^2. Furthermore, f\colon I \to (0, \infty) is logarithmically convex if and only if e^f(x) is convex for all \alpha\in\mathbb R..


Sufficient conditions

If f_1, \ldots, f_n are logarithmically convex, and if w_1, \ldots, w_n are non-negative real numbers, then f_1^ \cdots f_n^ is logarithmically convex. If \_ is any family of logarithmically convex functions, then g = \sup_ f_i is logarithmically convex. If f \colon X \to I \subseteq \mathbf is convex and g \colon I \to \mathbf_ is logarithmically convex and non-decreasing, then g \circ f is logarithmically convex.


Properties

A logarithmically convex function ''f'' is a convex function since it is the
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
of the
increasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
convex function \exp and the function \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 f(x) = x^2 is convex, but its logarithm \log f(x) = 2\log , x, is not. Therefore the squaring function is not logarithmically convex.


Examples

* f(x) = \exp(, x, ^p) is logarithmically convex when p \ge 1 and strictly logarithmically convex when p > 1. * f(x) = \frac is strictly logarithmically convex on (0,\infty) for all p>0. * Euler's
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
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 In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
function to real arguments.


See also

* Logarithmically concave function


Notes


References

* John B. Conway. ''Functions of One Complex Variable I'', second edition. Springer-Verlag, 1995. . * * . * . {{PlanetMath attribution, id=5664, title=logarithmically convex function Real analysis