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In `F33f`_`[convex analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_analysis]`_`f, a `F33f`_`[non-negative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Non-negative]`_`f function `*f`* : `!R`!`*n`* → `!R`!+ is `!logarithmically concave`! (or `!log-concave`! for short) if its `F33f`_`[domain`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Domain_of_a_function]`_`f is a `F33f`_`[convex set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_set]`_`f, and if it satisfies the inequality

f ( θ θ x + ( 1 − − θ θ ) y ) ≥ ≥ f ( x ) θ θ f ( y ) 1 − − θ θ {\\displaystyle f(\\theta x+(1-\\theta )y)\\geq f(x)^{\\theta }f(y)^{1-\\theta }}

for all `*x`*,`*y`* ∈ dom `*f`* and 0 < `*θ`* < 1. If `*f`* is strictly positive, this is equivalent to saying that the `F33f`_`[logarithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithm]`_`f of the function, log ∘ `*f`*, is `F33f`_`[concave`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Concave_function]`_`f; that is,

log ⁡ ⁡ f ( θ θ x + ( 1 − − θ θ ) y ) ≥ ≥ θ θ log ⁡ ⁡ f ( x ) + ( 1 − − θ θ ) log ⁡ ⁡ f ( y ) {\\displaystyle \\log f(\\theta x+(1-\\theta )y)\\geq \\theta \\log f(x)+(1-\\theta )\\log f(y)}

for all `*x`*,`*y`* ∈ dom `*f`* and 0 < `*θ`* < 1.

Examples of log-concave functions are the 0-1 `F33f`_`[indicator functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Indicator_function]`_`f of convex sets (which requires the more flexible definition), and the `F33f`_`[Gaussian function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gaussian_function]`_`f.

Similarly, a function is `*`F33f`_`[log-convex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Log-convex]`_`f`* if it satisfies the reverse inequality

f ( θ θ x + ( 1 − − θ θ ) y ) ≤ ≤ f ( x ) θ θ f ( y ) 1 − − θ θ {\\displaystyle f(\\theta x+(1-\\theta )y)\\leq f(x)^{\\theta }f(y)^{1-\\theta }}

for all `*x`*,`*y`* ∈ dom `*f`* and 0 < `*θ`* < 1.

>>Contents

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

• `F0af`_`[Operations preserving log-concavity`#operations-preserving-log-concavity]`_`f

• `F0af`_`[Log-concave distributions`#log-concave-distributions]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[Notes`#notes]`_`f

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

-─

>>Properties

• A log-concave function is also `F33f`_`[quasi-concave`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quasi-concave_function]`_`f. This follows from the fact that the logarithm is monotone implying that the `F33f`_`[superlevel sets`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Level_set]`_`f of this function are convex.`:cite-ref-0-1-0[`F5bf`_`[1`#cite-note-0-1]`_`f]

• Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the `F33f`_`[Gaussian function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gaussian_function]`_`f `*f`*(`*x`*) = exp(−`*x`*2/2) which is log-concave since log `*f`*(`*x`*) = −`*x`*2/2 is a concave function of `*x`*. But `*f`* is not concave since the `F33f`_`[second derivative`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Second_derivative]`_`f is positive for |`*x`*| > 1:

f ″ ( x ) = e − − x 2 2 ( x 2 − − 1 ) ≰ ≰ 0 {\\displaystyle f''(x)=e^{-{\\frac {x^{2}}{2}}}(x^{2}-1)\\nleq 0}

• From above two points, `F33f`_`[concavity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Concave_function]`_`f ⇒ ⇒ {\\displaystyle \\Rightarrow } log-concavity ⇒ ⇒ {\\displaystyle \\Rightarrow } `F33f`_`[quasiconcavity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quasiconcave_function]`_`f.

• A twice differentiable, nonnegative function with a convex domain is log-concave `F33f`_`[if and only if`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=If_and_only_if]`_`f for all `*x`* satisfying `*f`*(`*x`*) > 0,

f ( x ) ∇ ∇ 2 f ( x ) ⪯ ⪯ ∇ ∇ f ( x ) ∇ ∇ f ( x ) T {\\displaystyle f(x)\\nabla ^{2}f(x)\\preceq \\nabla f(x)\\nabla f(x)^{T}} ,`:cite-ref-0-1-1[`F5bf`_`[1`#cite-note-0-1]`_`f]

i.e.

f ( x ) ∇ ∇ 2 f ( x ) − − ∇ ∇ f ( x ) ∇ ∇ f ( x ) T {\\displaystyle f(x)\\nabla ^{2}f(x)-\\nabla f(x)\\nabla f(x)^{T}} is

`F33f`_`[negative semi-definite`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Positive-definite_matrix]`_`f. For functions of one variable, this condition simplifies to

f ( x ) f ″ ( x ) ≤ ≤ ( f ′ ( x ) ) 2 {\\displaystyle f(x)f''(x)\\leq (f'(x))^{2}}

>>Operations preserving log-concavity

• Products: The product of log-concave functions is also log-concave. Indeed, if `*f`* and `*g`* are log-concave functions, then log `*f`* and log `*g`* are concave by definition. Therefore

log f ( x ) + log g ( x ) = log ⁡ ⁡ ( f ( x ) g ( x ) ) {\\displaystyle \\log \\,f(x)+\\log \\,g(x)=\\log(f(x)g(x))}

is concave, and hence also `*f`* `*g`* is log-concave.

• `F33f`_`[Marginals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Marginal_distribution]`_`f: if `*f`*(`*x`*,`*y`*) : `!R`!`*n`*+`*m`* → `!R`! is log-concave, then

g ( x ) = ∫ ∫ f ( x , y ) d y {\\displaystyle g(x)=\\int f(x,y)dy}

is log-concave (see `F33f`_`[Prékopa–Leindler inequality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Prékopa–Leindler_inequality]`_`f).

• This implies that `F33f`_`[convolution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convolution]`_`f preserves log-concavity, since `*h`*(`*x`*,`*y`*) = `*f`*(`*x`*-`*y`*) `*g`*(`*y`*) is log-concave if `*f`* and `*g`* are log-concave, and therefore

( f ∗ ∗ g ) ( x ) = ∫ ∫ f ( x − − y ) g ( y ) d y = ∫ ∫ h ( x , y ) d y {\\displaystyle (f*g)(x)=\\int f(x-y)g(y)dy=\\int h(x,y)dy}

is log-concave.

>>Log-concave distributions

Log-concave distributions are necessary for a number of algorithms, e.g. `F33f`_`[adaptive rejection sampling`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Adaptive_rejection_sampling]`_`f. Every distribution with log-concave density is a `F33f`_`[maximum entropy probability distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Maximum_entropy_probability_distribution]`_`f with specified mean `*μ`* and `F33f`_`[Deviation risk measure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Deviation_risk_measure]`_`f `*D`*.`:cite-ref-grechuk1-2-0[`F5bf`_`[2`#cite-note-grechuk1-2]`_`f] As it happens, many common `F33f`_`[probability distributions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_distribution]`_`f are log-concave. Some examples:`:cite-ref-1-3-0[`F5bf`_`[3`#cite-note-1-3]`_`f]

• the `F33f`_`[normal distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal_distribution]`_`f and `F33f`_`[multivariate normal distributions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Multivariate_normal_distribution]`_`f,

• the `F33f`_`[exponential distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exponential_distribution]`_`f,

• the `F33f`_`[uniform distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Uniform_distribution_(continuous)]`_`f over any `F33f`_`[convex set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convex_set]`_`f,

• the `F33f`_`[binomial distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Binomial_distribution]`_`f,

• the `F33f`_`[logistic distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logistic_distribution]`_`f,

• the `F33f`_`[extreme value distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Extreme_value_distribution]`_`f,

• the `F33f`_`[Laplace distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Laplace_distribution]`_`f,

• the `F33f`_`[chi distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chi_distribution]`_`f,

• the `F33f`_`[hyperbolic secant distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hyperbolic_secant_distribution]`_`f,

• the `F33f`_`[Wishart distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Wishart_distribution]`_`f, if `*n`* ≥ `*p`* + 1,`:cite-ref-prekopa-4-0[`F5bf`_`[4`#cite-note-prekopa-4]`_`f]

• the `F33f`_`[Dirichlet distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirichlet_distribution]`_`f, if all parameters are ≥ 1,`:cite-ref-prekopa-4-1[`F5bf`_`[4`#cite-note-prekopa-4]`_`f]

• the `F33f`_`[gamma distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gamma_distribution]`_`f if the `F33f`_`[shape parameter`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Shape_parameter]`_`f is ≥ 1,

• the `F33f`_`[chi-square distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Chi-square_distribution]`_`f if the number of degrees of freedom is ≥ 2,

• the `F33f`_`[beta distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Beta_distribution]`_`f if both shape parameters are ≥ 1, and

• the `F33f`_`[Weibull distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weibull_distribution]`_`f if the shape parameter is ≥ 1.

Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave.

The following distributions are non-log-concave for all parameters:

• the `F33f`_`[Student's t-distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Student's_t-distribution]`_`f,

• the `F33f`_`[Cauchy distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cauchy_distribution]`_`f,

• the `F33f`_`[Pareto distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pareto_distribution]`_`f,

• the `F33f`_`[log-normal distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Log-normal_distribution]`_`f, and

• the `F33f`_`[F-distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=F-distribution]`_`f.

Note that the `F33f`_`[cumulative distribution function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cumulative_distribution_function]`_`f (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's:

• the `F33f`_`[log-normal distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Log-normal_distribution]`_`f,

• the `F33f`_`[Pareto distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Pareto_distribution]`_`f,

• the `F33f`_`[Weibull distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Weibull_distribution]`_`f when the shape parameter < 1, and

• the `F33f`_`[gamma distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gamma_distribution]`_`f when the shape parameter < 1.

The following are among the properties of log-concave distributions:

• If a density is log-concave, so is its `F33f`_`[cumulative distribution function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cumulative_distribution_function]`_`f (CDF).

• If a multivariate density is log-concave, so is the `F33f`_`[marginal density`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Marginal_density]`_`f over any subset of variables.

• The sum of two independent log-concave `F33f`_`[random variables`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Random_variable]`_`f is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.

• The product of two log-concave functions is log-concave. This means that `F33f`_`[joint`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Joint_distribution]`_`f densities formed by multiplying two probability densities (e.g. the `F33f`_`[normal-gamma distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Normal-gamma_distribution]`_`f, which always has a shape parameter ≥ 1) will be log-concave. This property is heavily used in general-purpose `F33f`_`[Gibbs sampling`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gibbs_sampling]`_`f programs such as `F33f`_`[BUGS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bayesian_inference_using_Gibbs_sampling]`_`f and `F33f`_`[JAGS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Just_another_Gibbs_sampler]`_`f, which are thereby able to use `F33f`_`[adaptive rejection sampling`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Adaptive_rejection_sampling]`_`f over a wide variety of `F33f`_`[conditional distributions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Conditional_distribution]`_`f derived from the product of other distributions.

• If a density is log-concave, so is its `F33f`_`[survival function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Survival_function]`_`f.`:cite-ref-1-3-1[`F5bf`_`[3`#cite-note-1-3]`_`f]

• If a density is log-concave, it has a monotone `F33f`_`[hazard rate`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hazard_rate]`_`f (MHR), and is a `F33f`_`[regular distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Regular_distribution_(economics)]`_`f since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.

d d x log ⁡ ⁡ ( 1 − − F ( x ) ) = − − f ( x ) 1 − − F ( x ) {\\displaystyle {\\frac {d}{dx}}\\log \\left(1-F(x)\\right)=-{\\frac {f(x)}{1-F(x)}}} which is decreasing as it is the derivative of a concave function.

>>See also

• `F33f`_`[logarithmically concave sequence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmically_concave_sequence]`_`f

• `F33f`_`[logarithmically concave measure`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmically_concave_measure]`_`f

• `F33f`_`[logarithmically convex function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmically_convex_function]`_`f

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

>>Notes

`:cite-note-0-1`!1.`! `F0af`_`[↑`#cite-ref-0-1-0]`_`f `:citerefboydvandenberghe2004`a`F33f`_`[Boyd, Stephen`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Stephen_P._Boyd]`_`f; Vandenberghe, Lieven (2004). "Log-concave and log-convex functions". `*Convex Optimization`*. Cambridge University Press. pp. 104–108. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-521-83378-7.

`:cite-note-grechuk1-2`!2.`! `F0af`_`[↑`#cite-ref-grechuk1-2-0]`_`f `:citerefgrechukmolybohazabarankin2009`aGrechuk, Bogdan; Molyboha, Anton; Zabarankin, Michael (May 2009). "Maximum Entropy Principle with General Deviation Measures" (PDF). `*`F33f`_`[Mathematics of Operations Research`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics_of_Operations_Research]`_`f`*. `!34`! (2): 445–467. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1287/moor.1090.0377.

`:cite-note-1-3`!3.`! `F0af`_`[↑`#cite-ref-1-3-0]`_`f See `:citerefbagnolibergstrom2005`aBagnoli, Mark; Bergstrom, Ted (2005). "Log-Concave Probability and Its Applications" (PDF). `*Economic Theory`*. `!26`! (2): 445–469. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/s00199-004-0514-4. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 1046688.

`:cite-note-prekopa-4`!4.`! `F0af`_`[↑`#cite-ref-prekopa-4-0]`_`f `:citerefpr-kopa1971`a`F33f`_`[Prékopa, András`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=András_Prékopa]`_`f (1971). "Logarithmic concave measures with application to stochastic programming" (PDF). `*`F33f`_`[Acta Scientiarum Mathematicarum`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Acta_Scientiarum_Mathematicarum]`_`f`*. `!32`! (3–4): 301–316.

>>References

• `:citerefbarndorff-nielsen1978`a`F33f`_`[Barndorff-Nielsen, Ole`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Ole_Barndorff-Nielsen]`_`f (1978). `*Information and exponential families in statistical theory`*. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley \\& Sons, Ltd. pp. ix+238 pp. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-471-99545-2. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0489333.

• `:citerefdharmadhikarijoag-dev1988`aDharmadhikari, Sudhakar; Joag-Dev, Kumar (1988). `*Unimodality, convexity, and applications`*. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-12-214690-5. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0954608.

• `:citerefpfanzaglwith-the-assistance-of-r-hamb-ker1994`aPfanzagl, Johann; with the assistance of R. Hamböker (1994). `*Parametric Statistical Theory`*. Walter de Gruyter. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 3-11-013863-8. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 1291393.

• `:citerefpe-ari-proschantong1992`aPečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992). `*Convex functions, partial orderings, and statistical applications`*. Mathematics in Science and Engineering. Vol. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-12-549250-2. `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 1162312.

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