Difference between revisions of "Convexity, logarithmic"
From Encyclopedia of Mathematics
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| − | The property of a non-negative function | + | The property of a non-negative function $f$, defined on some interval, that can be described as follows: If for any $x_1$ and $x_2$ in this interval and for any $p_1 \ge 0$, $p_2 \ge 0$ with $p_1+p_2=1$ the inequality |
| + | $$ | ||
| + | f(p_1x_1 + p_2x_2) \le f(x_1)^{p_1} \cdot f(x_2)^{p_2} | ||
| + | $$ | ||
| + | is satisfied, $f$ is called logarithmically convex. If a function is logarithmically convex, it is either identically equal to zero or is strictly positive and $\log f$ is a [[convex function (of a real variable)]]. | ||
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| − | + | {{TEX|done}} | |
Revision as of 15:40, 29 October 2016
The property of a non-negative function $f$, defined on some interval, that can be described as follows: If for any $x_1$ and $x_2$ in this interval and for any $p_1 \ge 0$, $p_2 \ge 0$ with $p_1+p_2=1$ the inequality $$ f(p_1x_1 + p_2x_2) \le f(x_1)^{p_1} \cdot f(x_2)^{p_2} $$ is satisfied, $f$ is called logarithmically convex. If a function is logarithmically convex, it is either identically equal to zero or is strictly positive and $\log f$ is a convex function (of a real variable).
How to Cite This Entry:
Convexity, logarithmic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convexity,_logarithmic&oldid=11737
Convexity, logarithmic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convexity,_logarithmic&oldid=11737
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article