logarithmically convex function

Definition.

A function $f\colon[a,b]\to{\mathbb{R}}$ such that $f(x)>0$ for all $x$ is said to be logarithmically convex if $\log f(x)$ 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 $f(x)=x^{2}$ is a convex function, but $\log f(x)=\log x^{2}=2\log x$ is not a convex function and thus $f(x)=x^{2}$ is not logarithmically convex. On the other hand $e^{x^{2}}$ is logarithmically convex since $\log e^{x^{2}}=x^{2}$ is convex. A less trivial example of a logarithmically convex function is the gamma function, if restricted to the positive reals.

The definition is easily extended to functions $f\colon U\subset{\mathbb{R}}\to{\mathbb{R}}$, 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 $[a,b]\subset U$.

References

• 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