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^[1] 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).

Properties

• Log-convexity $\Rightarrow$ convexity $\Rightarrow$ quasiconvexity.^[2]

References

1. Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.
2. Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. ISBN 9780521833783.

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

See also

• logarithmically concave function

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