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logarithmically convex function (Definition)
Definition 1   A function $ f\colon [a,b] \to {\mathbb{R}}$ such that $ f(x) > 0$ for all $ x$ is said to be logarithmically convex if $ \log f(x)$ is a convex function.

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.

The definition is easily extended to functions $ f\colon U \subset {\mathbb{R}} \to {\mathbb{R}}$, for any connected set $ U$ (where still we have $ f > 0$), in the obvious way. Such a function is logarithmically convex if it is logarithmically convex on all intervals $ [a,b] \subset U$.

Bibliography

1
John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.



"logarithmically convex function" is owned by jirka.
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See Also: convex function, Bohr-Mollerup theorem, Hadamard three-circle theorem

Other names:  logarithmically convex, log-convex function, log-convex, log convex function, log convex
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Cross-references: intervals, connected, reals, positive, restricted, gamma function, convex, converse, easy to see, convex function, function
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This is version 4 of logarithmically convex function, born on 2004-03-02, modified 2006-08-04.
Object id is 5664, canonical name is LogarithmicallyConvexFunction.
Accessed 8124 times total.

Classification:
AMS MSC: 26A51 (Real functions :: Functions of one variable :: Convexity, generalizations)
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