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
∘
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
∘
f
, which is supposed convex. The converse is not always true: for example
g
:
x
↦
x
2
is a convex function, but
log
∘
g
:
x
↦
log
x
2
=
2
log
|
x
|
is not a convex function and thus
g
is not logarithmically convex. On the other hand,
x
↦
e
x
2
is logarithmically convex since
x
↦
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).
