logarithmically convex function


A functionMathworldPlanetmath f:[a,b] such that f(x)>0 for all x is said to be logarithmically convex if logf(x) is a convex function.

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 ex2 is logarithmically convex since logex2=x2 is convex. A less trivial example of a logarithmically convex function is the gamma functionDlmfDlmfMathworldPlanetmath, if restricted to the positive reals.

The definition is easily extended to functions f:U, for any connected set U (where still we have f>0), in the obvious way. Such a function is logarithmically convex if it is logarithmically convex on all intervals [a,b]U.


  • 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title logarithmically convex function
Canonical name LogarithmicallyConvexFunction
Date of creation 2013-03-22 14:13:33
Last modified on 2013-03-22 14:13:33
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Definition
Classification msc 26A51
Synonym logarithmically convex
Synonym log-convex function
Synonym log-convex
Synonym log convex function
Synonym log convex
Related topic ConvexFunction
Related topic BohrMollerupTheorem
Related topic HadamardThreeCircleTheorem