# Logarithmically convex function

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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 ${\displaystyle {\log }\circ f}$, 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 ${\displaystyle f}$, so if the composition still retains the convexity property, this must mean that the original function ${\displaystyle f}$ 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 ${\displaystyle \exp }$ and the function ${\displaystyle \log \circ f}$, which is supposed convex. The converse is not always true: for example ${\displaystyle g:x\mapsto x^{2}}$ is a convex function, but ${\displaystyle {\log }\circ g:x\mapsto \log x^{2}=2\log |x|}$ is not a convex function and thus ${\displaystyle g}$ is not logarithmically convex. On the other hand, ${\displaystyle x\mapsto e^{x^{2}}}$ is logarithmically convex since ${\displaystyle x\mapsto \log e^{x^{2}}=x^{2}}$ 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 ${\displaystyle \Rightarrow }$ convexity ${\displaystyle \Rightarrow }$ 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.