Logarithmically convex function

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In mathematics, a function <math>f</math> defined on an open interval of the real line with positive values is said to be logarithmically convex if <math>\log f(x)</math> is a convex function of <math>x</math>.

It is easy to see that a logarithmically convex function is a convex function, but the converse is not true. For example <math>f(x) = x^2</math> is a convex function, but <math>\log f(x) = \log x^2 = 2 \log x</math> is not a convex function and thus <math>f(x) = x^2</math> is not logarithmically convex. On the other hand, <math>f(x)=e^{x^2}</math> is logarithmically convex since <math>\log e^{x^2} = x^2</math> is convex. A less trivial example of a logarithmically convex function is the gamma function, if restricted to the positive reals (see also the Bohr-Mollerup theorem).