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logarithmically convex function
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(Definition)
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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 $[a,b] \subset U$ .
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- John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
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"logarithmically convex function" is owned by jirka.
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Cross-references: intervals, connected, reals, positive, gamma function, convex, converse, easy to see, convex function, function
There are 4 references to this entry.
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 10248 times total.
Classification:
| AMS MSC: | 26A51 (Real functions :: Functions of one variable :: Convexity, generalizations) |
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Pending Errata and Addenda
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