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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, a `F33f`_`[function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f `*f`* is `!logarithmically convex`! or `!superconvex`!`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] if log ∘ ∘ f {\\displaystyle {\\log }\\circ f} , the `F33f`_`[composition`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_composition]`_`f of the `F33f`_`[logarithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithm]`_`f with `*f`*, is itself a `F33f`_`[convex function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_function]`_`f.

>>Contents

• `F0af`_`[Definition`#definition]`_`f

• `F0af`_`[Equivalent conditions`#equivalent-conditions]`_`f

• `F0af`_`[Sufficient conditions`#sufficient-conditions]`_`f

• `F0af`_`[Properties`#properties]`_`f

• `F0af`_`[Examples`#examples]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>Definition

Let `*X`* be a `F33f`_`[convex subset`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_set]`_`f of a `F33f`_`[real`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_numbers]`_`f `F33f`_`[vector space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_space]`_`f, and let `*f`* : `*X`* → `!R`! be a function taking `F33f`_`[non-negative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Negative_and_positive_numbers]`_`f values. Then `*f`* is:

• `!Logarithmically convex`! if log ∘ ∘ f {\\displaystyle {\\log }\\circ f} is convex, and

• `!Strictly logarithmically convex`! if log ∘ ∘ f {\\displaystyle {\\log }\\circ f} is strictly convex.

Here we interpret log ⁡ ⁡ 0 {\\displaystyle \\log 0} as − − ∞ ∞ {\\displaystyle -\\infty } .

Explicitly, `*f`* is logarithmically convex if and only if, for all `*x`*1, `*x`*2 ∈ `*X`* and all `*t`* ∈ [0, 1], the two following equivalent conditions hold:

log ⁡ ⁡ f ( t x 1 + ( 1 − − t ) x 2 ) ≤ ≤ t log ⁡ ⁡ f ( x 1 ) + ( 1 − − t ) log ⁡ ⁡ f ( x 2 ) , f ( t x 1 + ( 1 − − t ) x 2 ) ≤ ≤ f ( x 1 ) t f ( x 2 ) 1 − − t . {\\displaystyle {\\begin{aligned}\\log f(tx_{1}+(1-t)x_{2})&\\leq t\\log f(x_{1})+(1-t)\\log f(x_{2}),\\\\f(tx_{1}+(1-t)x_{2})&\\leq f(x_{1})^{t}f(x_{2})^{1-t}.\\end{aligned}}}

Similarly, `*f`* is strictly logarithmically convex if and only if, in the above two expressions, strict inequality holds for all `*t`* ∈ (0, 1).

The above definition permits `*f`* to be zero, but if `*f`* is logarithmically convex and vanishes anywhere in `*X`*, then it vanishes everywhere in the interior of `*X`*.

>>>Equivalent conditions

If `*f`* is a `F33f`_`[differentiable function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Differentiable_function]`_`f defined on an interval `*I`* ⊆ `!R`!, then `*f`* is logarithmically convex if and only if the following condition holds for all `*x`* and `*y`* in `*I`*:

log ⁡ ⁡ f ( x ) ≥ ≥ log ⁡ ⁡ f ( y ) + f ′ ( y ) f ( y ) ( x − − y ) . {\\displaystyle \\log f(x)\\geq \\log f(y)+{\\frac {f'(y)}{f(y)}}(x-y).}

This is equivalent to the condition that, whenever `*x`* and `*y`* are in `*I`* and `*x`* > `*y`*,

( f ( x ) f ( y ) ) 1 x − − y ≥ ≥ exp ⁡ ⁡ ( f ′ ( y ) f ( y ) ) . {\\displaystyle \\left({\\frac {f(x)}{f(y)}}\\right)^{\\frac {1}{x-y}}\\geq \\exp \\left({\\frac {f'(y)}{f(y)}}\\right).}

Moreover, `*f`* is strictly logarithmically convex if and only if these inequalities are always strict.

If `*f`* is twice differentiable, then it is logarithmically convex if and only if, for all `*x`* in `*I`*,

f ″ ( x ) f ( x ) ≥ ≥ f ′ ( x ) 2 . {\\displaystyle f''(x)f(x)\\geq f'(x)^{2}.}

If the inequality is always strict, then `*f`* is strictly logarithmically convex. However, the converse is false: It is possible that `*f`* is strictly logarithmically convex and that, for some `*x`*, we have f ″ ( x ) f ( x ) = f ′ ( x ) 2 {\\displaystyle f''(x)f(x)=f'(x)^{2}} . For example, if f ( x ) = exp ⁡ ⁡ ( x 4 ) {\\displaystyle f(x)=\\exp(x^{4})} , then `*f`* is strictly logarithmically convex, but f ″ ( 0 ) f ( 0 ) = 0 = f ′ ( 0 ) 2 {\\displaystyle f''(0)f(0)=0=f'(0)^{2}} .

Furthermore, f : : I → → ( 0 , ∞ ∞ ) {\\displaystyle f\\colon I\\to (0,\\infty )} is logarithmically convex if and only if e α α x f ( x ) {\\displaystyle e^{\\alpha x}f(x)} is convex for all α α ∈ ∈ R {\\displaystyle \\alpha \\in \\mathbb {R} } .`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f]

>>Sufficient conditions

If f 1 , … … , f n {\\displaystyle f_{1},\\ldots ,f_{n}} are logarithmically convex, and if w 1 , … … , w n {\\displaystyle w_{1},\\ldots ,w_{n}} are non-negative real numbers, then f 1 w 1 ⋯ ⋯ f n w n {\\displaystyle f_{1}^{w_{1}}\\cdots f_{n}^{w_{n}}} is logarithmically convex.

If { f i } i ∈ ∈ I {\\displaystyle \\{f_{i}\\}_{i\\in I}} is any family of logarithmically convex functions, then g = sup i ∈ ∈ I f i {\\displaystyle g=\\sup _{i\\in I}f_{i}} is logarithmically convex.

If f : : X → → I ⊆ ⊆ R {\\displaystyle f\\colon X\\to I\\subseteq \\mathbf {R} } is convex and g : : I → → R ≥ ≥ 0 {\\displaystyle g\\colon I\\to \\mathbf {R} _{\\geq 0}} is logarithmically convex and non-decreasing, then g ∘ ∘ f {\\displaystyle g\\circ f} is logarithmically convex.

>>Properties

A logarithmically convex function `*f`* is a convex function since it is the `F33f`_`[composite`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_composition]`_`f of the `F33f`_`[increasing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Increasing_function]`_`f convex function exp {\\displaystyle \\exp } and the function log ∘ ∘ f {\\displaystyle \\log \\circ f} , which is by definition convex. However, being logarithmically convex is a strictly stronger property than being convex. For example, the squaring function f ( x ) = x 2 {\\displaystyle f(x)=x^{2}} is convex, but its logarithm log ⁡ ⁡ f ( x ) = 2 log ⁡ ⁡ | x | {\\displaystyle \\log f(x)=2\\log |x|} is not. Therefore the squaring function is not logarithmically convex.

>>Examples

• f ( x ) = exp ⁡ ⁡ ( | x | p ) {\\displaystyle f(x)=\\exp(|x|^{p})} is logarithmically convex when p ≥ ≥ 1 {\\displaystyle p\\geq 1} and strictly logarithmically convex when p > 1 {\\displaystyle p>1} .

• f ( x ) = 1 x p {\\displaystyle f(x)={\\frac {1}{x^{p}}}} is strictly logarithmically convex on ( 0 , ∞ ∞ ) {\\displaystyle (0,\\infty )} for all p > 0. {\\displaystyle p>0.}

• Euler's `F33f`_`[gamma function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gamma_function]`_`f is strictly logarithmically convex when restricted to the positive real numbers. In fact, by the `F33f`_`[Bohr–Mollerup theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bohr–Mollerup_theorem]`_`f, this property can be used to characterize Euler's gamma function among the possible extensions of the `F33f`_`[factorial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Factorial]`_`f function to real arguments.

>>See also

• `F33f`_`[Logarithmically concave function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmically_concave_function]`_`f

>>Notes

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `F33f`_`[Montel 1928`#citerefmontel1928]`_`f.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `F33f`_`[NiculescuPersson 2006`#citerefniculescupersson2006]`_`f, p. 70.

>>References

• John B. Conway. `*Functions of One Complex Variable I`*, second edition. Springer-Verlag, 1995. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-387-90328-3.

• "Convexity, logarithmic", `*`F33f`_`[Encyclopedia of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Encyclopedia_of_Mathematics]`_`f`*, `F33f`_`[EMS Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=European_Mathematical_Society]`_`f, 2001 [1994]

• `:citerefniculescupersson2006`aNiculescu, Constantin; `F33f`_`[Persson, Lars-Erik`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lars-Erik_Persson]`_`f (2006), `*Convex Functions and their Applications - A Contemporary Approach`* (1st ed.), `F33f`_`[Springer`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Springer-Verlag]`_`f, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/0-387-31077-0, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-387-24300-9, `F33f`_`[ISSN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISSN_(identifier)]`_`f 1613-5237.

• `:citerefmontel1928`a`F33f`_`[Montel, Paul`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Paul_Montel]`_`f (1928), "Sur les fonctions convexes et les fonctions sousharmoniques", `*Journal de Mathématiques Pures et Appliquées`* (in French), `!7`!: 29–60.

`*This article incorporates material from logarithmically convex function on `F33f`_`[PlanetMath`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=PlanetMath]`_`f, which is licensed under the Creative Commons Attribution/Share-Alike License.`*

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