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Difference between revisions of "Convexity, logarithmic"

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The property of a non-negative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264101.png" />, defined on some interval, that can be described as follows: If for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264103.png" /> in this interval and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264105.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264106.png" /> the inequality
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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
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$$
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f(p_1x_1 + p_2x_2) \le f(x_1)^{p_1} \cdot f(x_2)^{p_2}
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$$
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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)]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264107.png" /></td> </tr></table>
 
  
is satisfied, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264108.png" /> is called logarithmically convex. If a function is logarithmically convex, it is either identically equal to zero or is strictly positive and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026410/c0264109.png" /> is a [[Convex function (of a real variable)|convex function (of a real variable)]].
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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=39512
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article