Logarithmically convex function

In mathematics, a function Failed to parse (Missing texvc executable; please see math/README to configure.): f

defined on an open interval of the real line with positive values is said to be logarithmically convex if Failed to parse (Missing texvc executable; please see math/README to configure.): \log f(x)
is a convex function of Failed to parse (Missing texvc executable; please see math/README to configure.): x

.

It is easy to see that a logarithmically convex function is a convex function, but the converse is not true. For example Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = x^2

is a convex function, but Failed to parse (Missing texvc executable; please see math/README to configure.): \log f(x) = \log x^2 = 2 \log x
is not a convex function and thus Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = x^2
is not logarithmically convex. On the other hand, Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)=e^{x^2}
is logarithmically convex since Failed to parse (Missing texvc executable; please see math/README to configure.): \log e^{x^2} = x^2
is convex. A less trivial example of a logarithmically convex function is the gamma function, if restricted to the positive reals (see also the Bohr-Mollerup theorem).

References

• John B. Conway. Functions of One Complex Variable I, second edition. Springer-Verlag, 1995. ISBN 0-387-90328-3.

• This article incorporates material from logarithmically convex function on PlanetMath, which is licensed under the GFDL.