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Homelogarithmically convex function

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# logarithmically convex function

###### Definition.

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 $[a,b]\subset U$.

# References

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

## Mathematics Subject Classification

26A51*no label found*

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