Logarithmically convex function
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In mathematics, a function f defined on an open interval of the real line with positive values is said to be logarithmically convex if logf(x) is a convex function of x.
It is easy to see that a logarithmically convex function is a convex function, but the converse is not true. For example f(x) = x2 is a convex function, but logf(x) = logx2 = 2logx is not a convex function and thus f(x) = x2 is not logarithmically convex. On the other hand, 
is logarithmically convex since 
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).
[edit] 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.
