Logarithmically convex function

In mathematics, a function f defined on a convex subset of a real vector space and taking positive values is said to be logarithmically convex or superconvex if ${\log}\circ f$, the composition of the logarithmic function with f, is a convex function. In effect the logarithm drastically slows down the growth of the original function $f$, so if the composition still retains the convexity property, this must mean that the original function $f$ was 'really convex' to begin with, hence the term superconvex.

A logarithmically convex function f is a convex function since it is the composite of the increasing convex function $\exp$ and the function $\log\circ f$, which is supposed convex. The converse is not always true: for example $g: x\mapsto x^2$ is a convex function, but ${\log}\circ g: x\mapsto \log x^2 = 2 \log |x|$ is not a convex function and thus $g$ is not logarithmically convex. On the other hand, $x\mapsto e^{x^2}$ is logarithmically convex since $x\mapsto \log e^{x^2} = x^2$ is convex. An important example of a logarithmically convex function is the gamma function on 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.
• Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. ISBN 9780521833783.

See also

• logarithmically concave function

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