↑ arvau

Wikipedia / Logarithmically convex function

In mathematics, a function f is logarithmically convex or superconvex, the composition of the logarithm with f, is itself a convex function.

Definition

Let be a convex subset of a real vector space, and let be a function taking non-negative values. Then is:

  • Logarithmically convex is convex, and

  • Strictly logarithmically convex is strictly convex.

.

Explicitly, is logarithmically convex if and only if, for all and all , the two following equivalent conditions hold:

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 :

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

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 ,

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 , then .

.

Sufficient conditions

is logarithmically convex.

is logarithmically convex.

is logarithmically convex.

Properties

A logarithmically convex function f is a convex function since it is the composite of the increasing is not. Therefore the squaring function is not logarithmically convex.

Examples

  • .

  • 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

Notes

References

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

  • .

  • .