Logarithmically convex function - Revision history

Citation bot: Alter: author2-link, author1-link. | Use this bot. Report bugs. | Suggested by Dominic3203 | Linked from User:Mathbot/Possible_redirects | #UCB_webform_linked 436/972

2024-12-13T05:19:55Z

Alter: author2-link, author1-link. | Use this bot. Report bugs. | Suggested by Dominic3203 | Linked from User:Mathbot/Possible_redirects | #UCB_webform_linked 436/972

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	\| last2 = Persson		\| last2 = Persson
	\| first2 = Lars-Erik		\| first2 = Lars-Erik
	\| author2-link = Lars-~~Erik_Persson~~		\| author2-link = Lars-Erik Persson
	\| title = Convex Functions and their Applications - A Contemporary Approach		\| title = Convex Functions and their Applications - A Contemporary Approach
	\| publisher = [[Springer-Verlag\|Springer]]		\| publisher = [[Springer-Verlag\|Springer]]
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	\| last1 = Montel		\| last1 = Montel
	\| first1 = Paul		\| first1 = Paul
	\| author1-link = ~~Paul_Montel~~		\| author1-link = Paul Montel
	\| title = Sur les fonctions convexes et les fonctions sousharmoniques		\| title = Sur les fonctions convexes et les fonctions sousharmoniques
	\| journal = Journal de Mathématiques Pures et Appliquées		\| journal = Journal de Mathématiques Pures et Appliquées

LucasBrown: Adding local short description: "Function whose composition with the logarithm is convex", overriding Wikidata description "function thats composition with the logarithm is a convex function"

2024-04-09T00:36:24Z

Adding local short description: "Function whose composition with the logarithm is convex", overriding Wikidata description "function thats composition with the logarithm is a convex function"

← Previous revision		Revision as of 00:36, 9 April 2024
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			{{Short description\|Function whose composition with the logarithm is convex}}
	In [[mathematics]], a [[function (mathematics)\|function]] ''f'' is '''logarithmically convex''' or '''superconvex'''<ref>Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.</ref> if <math>{\log}\circ f</math>, the [[function composition\|composition]] of the [[logarithm]] with ''f'', is itself a [[convex function]].		In [[mathematics]], a [[function (mathematics)\|function]] ''f'' is '''logarithmically convex''' or '''superconvex'''<ref>Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.</ref> if <math>{\log}\circ f</math>, the [[function composition\|composition]] of the [[logarithm]] with ''f'', is itself a [[convex function]].

Mazewaxie: WP:GENFIXES

2024-03-13T22:15:15Z

WP:GENFIXES

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	\| year = 1928		\| year = 1928
	\| language = French		\| language = French
	\| pages = ~~29-60~~		\| pages = 29–60
	\| volume = 7		\| volume = 7
	}}.		}}.

Mgkrupa: /* References */

2022-11-02T22:30:44Z

References

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	}}.		}}.

			{{Convex analysis and variational analysis}}
	{{ConvexAnalysis}}

	{{PlanetMath attribution\|id=5664\|title=logarithmically convex function}}		{{PlanetMath attribution\|id=5664\|title=logarithmically convex function}}

Mgkrupa: Added {{ConvexAnalysis}}

2021-02-02T02:25:41Z

Added {{ConvexAnalysis}}

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	\| volume = 7		\| volume = 7
	}}.		}}.

			{{ConvexAnalysis}}

	{{PlanetMath attribution\|id=5664\|title=logarithmically convex function}}		{{PlanetMath attribution\|id=5664\|title=logarithmically convex function}}

Mgkrupa: Moved the See also section into its correct location

2021-01-17T02:34:35Z

Moved the See also section into its correct location

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	* <math>f(x) = \frac{1}{x^p}</math> is strictly logarithmically convex on <math>(0,\infty)</math> for all <math>p>0.</math>		* <math>f(x) = \frac{1}{x^p}</math> is strictly logarithmically convex on <math>(0,\infty)</math> for all <math>p>0.</math>
	* Euler's [[gamma function]] is strictly logarithmically convex when restricted to the positive real numbers. In fact, by the [[Bohr–Mollerup theorem]], this property can be used to characterize Euler's gamma function among the possible extensions of the [[factorial]] function to real arguments.		* Euler's [[gamma function]] is strictly logarithmically convex when restricted to the positive real numbers. In fact, by the [[Bohr–Mollerup theorem]], this property can be used to characterize Euler's gamma function among the possible extensions of the [[factorial]] function to real arguments.

		⚫	==See also==
		⚫	* [[Logarithmically concave function]]

	==Notes==		==Notes==
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	\| volume = 7		\| volume = 7
	}}.		}}.

⚫	==See also==
⚫	* [[Logarithmically concave function]]

	{{PlanetMath attribution\|id=5664\|title=logarithmically convex function}}		{{PlanetMath attribution\|id=5664\|title=logarithmically convex function}}

Mgkrupa: Added section: "Sufficient conditions"

2021-01-17T02:32:47Z

Added section: "Sufficient conditions"

← Previous revision		Revision as of 02:32, 17 January 2021
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	Furthermore, <math>f\colon I \to (0, \infty)</math> is logarithmically convex if and only if <math>e^{\alpha x}f(x)</math> is convex for all <math>\alpha\in\mathbb R</math>.<ref>{{harvnb\|Montel\|1928}}.</ref><ref>{{harvnb\|NiculescuPersson\|2006\|p=70}}.</ref>		Furthermore, <math>f\colon I \to (0, \infty)</math> is logarithmically convex if and only if <math>e^{\alpha x}f(x)</math> is convex for all <math>\alpha\in\mathbb R</math>.<ref>{{harvnb\|Montel\|1928}}.</ref><ref>{{harvnb\|NiculescuPersson\|2006\|p=70}}.</ref>

			==Sufficient conditions==
⚫	==Properties==
⚫	A logarithmically convex function ''f'' is a convex function since it is the [[function composition\|composite]] of the [[increasing function\|increasing]] convex function <math>\exp</math> and the function <math>\log\circ f</math>, which is by definition convex. However, being logarithmically convex is a strictly stronger property than being convex. For example, the squaring function <math>f(x) = x^2</math> is convex, but its logarithm <math>\log f(x) = 2\log \|x\|</math> is not. Therefore the squaring function is not logarithmically convex.

	If <math>f_1, \ldots, f_n</math> are logarithmically convex, and if <math>w_1, \ldots, w_n</math> are non-negative real numbers, then <math>f_1^{w_1} \cdots f_n^{w_n}</math> is logarithmically convex.		If <math>f_1, \ldots, f_n</math> are logarithmically convex, and if <math>w_1, \ldots, w_n</math> are non-negative real numbers, then <math>f_1^{w_1} \cdots f_n^{w_n}</math> is logarithmically convex.

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	If <math>f \colon X \to I \subseteq \mathbf{R}</math> is convex and <math>g \colon I \to \mathbf{R}_{\ge 0}</math> is logarithmically convex and non-decreasing, then <math>g \circ f</math> is logarithmically convex.		If <math>f \colon X \to I \subseteq \mathbf{R}</math> is convex and <math>g \colon I \to \mathbf{R}_{\ge 0}</math> is logarithmically convex and non-decreasing, then <math>g \circ f</math> is logarithmically convex.

		⚫	==Properties==
		⚫	A logarithmically convex function ''f'' is a convex function since it is the [[function composition\|composite]] of the [[increasing function\|increasing]] convex function <math>\exp</math> and the function <math>\log\circ f</math>, which is by definition convex. However, being logarithmically convex is a strictly stronger property than being convex. For example, the squaring function <math>f(x) = x^2</math> is convex, but its logarithm <math>\log f(x) = 2\log \|x\|</math> is not. Therefore the squaring function is not logarithmically convex.

	==Examples==		==Examples==

Thatsme314: /* Examples */

2020-11-08T01:24:28Z

Examples

← Previous revision		Revision as of 01:24, 8 November 2020
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	==Examples==		==Examples==
	* <math>f(x) = \exp(\|x\|^p)</math> is logarithmically convex when <math>p \ge 1</math> and strictly logarithmically convex when <math>p > 1</math>.		* <math>f(x) = \exp(\|x\|^p)</math> is logarithmically convex when <math>p \ge 1</math> and strictly logarithmically convex when <math>p > 1</math>.
	* <math>f(x) = \frac{1}{x^p}</math> is strictly logarithmically convex on <math>(0,\infty)</math> for all <math>p>0</math>		* <math>f(x) = \frac{1}{x^p}</math> is strictly logarithmically convex on <math>(0,\infty)</math> for all <math>p>0.</math>
	* Euler's [[gamma function]] is strictly logarithmically convex when restricted to the positive real numbers. In fact, by the [[Bohr–Mollerup theorem]], this property can be used to characterize Euler's gamma function among the possible extensions of the [[factorial]] function to real arguments.		* Euler's [[gamma function]] is strictly logarithmically convex when restricted to the positive real numbers. In fact, by the [[Bohr–Mollerup theorem]], this property can be used to characterize Euler's gamma function among the possible extensions of the [[factorial]] function to real arguments.

Thatsme314: /* Equivalent conditions */

2020-11-08T01:20:04Z

Equivalent conditions

← Previous revision		Revision as of 01:20, 8 November 2020
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	If the inequality is always strict, then {{math\|''f''}} is strictly logarithmically convex. However, the converse is false: It is possible that {{math\|''f''}} is strictly logarithmically convex and that, for some {{math\|''x''}}, we have <math>f''(x)f(x) = f'(x)^2</math>. For example, if <math>f(x) = \exp(x^4)</math>, then {{math\|''f''}} is strictly logarithmically convex, but <math>f''(0)f(0) = 0 = f'(0)^2</math>.		If the inequality is always strict, then {{math\|''f''}} is strictly logarithmically convex. However, the converse is false: It is possible that {{math\|''f''}} is strictly logarithmically convex and that, for some {{math\|''x''}}, we have <math>f''(x)f(x) = f'(x)^2</math>. For example, if <math>f(x) = \exp(x^4)</math>, then {{math\|''f''}} is strictly logarithmically convex, but <math>f''(0)f(0) = 0 = f'(0)^2</math>.

	Furthermore, <math>f\colon I \to (0, \infty)</math> is logarithmically convex if and only if <math>e^{\alpha x}f(x)</math> is convex for all <math>\alpha\in\mathbb R</math>.<ref>{{harvnb\|Montel\|1928}}.</ref><ref>{{harvnb\|~~Niculescu Persson~~\|2006\|p=70}}.</ref>		Furthermore, <math>f\colon I \to (0, \infty)</math> is logarithmically convex if and only if <math>e^{\alpha x}f(x)</math> is convex for all <math>\alpha\in\mathbb R</math>.<ref>{{harvnb\|Montel\|1928}}.</ref><ref>{{harvnb\|NiculescuPersson\|2006\|p=70}}.</ref>

	==Properties==		==Properties==

Thatsme314: /* Equivalent conditions */

2020-11-08T01:19:53Z

Equivalent conditions

← Previous revision		Revision as of 01:19, 8 November 2020
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	If the inequality is always strict, then {{math\|''f''}} is strictly logarithmically convex. However, the converse is false: It is possible that {{math\|''f''}} is strictly logarithmically convex and that, for some {{math\|''x''}}, we have <math>f''(x)f(x) = f'(x)^2</math>. For example, if <math>f(x) = \exp(x^4)</math>, then {{math\|''f''}} is strictly logarithmically convex, but <math>f''(0)f(0) = 0 = f'(0)^2</math>.		If the inequality is always strict, then {{math\|''f''}} is strictly logarithmically convex. However, the converse is false: It is possible that {{math\|''f''}} is strictly logarithmically convex and that, for some {{math\|''x''}}, we have <math>f''(x)f(x) = f'(x)^2</math>. For example, if <math>f(x) = \exp(x^4)</math>, then {{math\|''f''}} is strictly logarithmically convex, but <math>f''(0)f(0) = 0 = f'(0)^2</math>.

	Furthermore, <math>f\colon I \to (0, \infty)</math> is logarithmically convex if and only if <math>e^{\alpha x}f(x)</math> is convex for all <math>\alpha\in\mathbb R</math>.<ref>{{harvnb\|Montel\|1928}}.</ref><ref>{{harvnb\|Niculescu-Persson\|2006\|p=70}}.</ref>		Furthermore, <math>f\colon I \to (0, \infty)</math> is logarithmically convex if and only if <math>e^{\alpha x}f(x)</math> is convex for all <math>\alpha\in\mathbb R</math>.<ref>{{harvnb\|Montel\|1928}}.</ref><ref>{{harvnb\|Niculescu Persson\|2006\|p=70}}.</ref>

	==Properties==		==Properties==