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Log-convex




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

The definition is easily extended to functions f\colon \R o \R where still we have f > 0, in the obvious way. Such a function is logarithmically convex if it is logarithmically convex on all intervals
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