Jensen's inequality

The Jensen's inequality is an elementary inequality for convex and concave functions . Because of its generality, it is the basis of many important inequalities, especially in analysis and information theory . The inequality is named after the Danish mathematician Johan Ludwig Jensen , who presented it on January 17, 1905 at a conference of the Danish Mathematical Society. Under slightly different conditions, it can be found at Otto Hölder's as early as 1889 .

Jensen's inequality states that the functional value of a convex function at a finite convex combination of support points is always less than or equal to a finite convex combination of the function values ​​of the support points. This means in particular that the weighted arithmetic mean of the function values ​​at n places is greater than or equal to the function value at the mean of these n places. Equality always applies to linear functions.

sentence

For a convex function and for nonnegative ones with : ${\displaystyle f\;}$ ${\displaystyle \lambda _{i}\;}$ ${\displaystyle \sum _{i=1}^{n}\lambda _{i}=1}$

${\displaystyle f\left(\sum _{i=1}^{n}\lambda _{i}x_{i}\right)\leq \sum _{i=1}^{n}\lambda _{i}f\left(x_{i}\right).}$

Proof by induction

If one uses the definition of convex that is common today, that

${\displaystyle f(\lambda x+(1-\lambda )y)\leq \lambda f(x)+(1-\lambda )f(y)}$

For all real values ​​between 0 and 1, Jensen's inequality results from complete induction over the number of support points. ${\displaystyle \lambda }$

Proof from Hölder

Hölder did not yet use the term convex and showed that from or monotonically increasing the inequality ${\displaystyle f''\geq 0}$ ${\displaystyle f'\;}$

${\displaystyle f\left({\frac {\sum _{i=1}^{n}a_{i}x_{i}}{\sum _{i=1}^{n}a_{i}}}\right)\leq {\frac {\sum _{i=1}^{n}a_{i}f\left(x_{i}\right)}{\sum _{i=1}^{n}a_{i}}}}$

for positive follows, whereby he essentially proved this with the mean value theorem of differential calculus . ${\displaystyle a_{i}\;}$

Proof from Jensen

Jensen went from the weaker definition

${\displaystyle f\left({\frac {x+y}{2}}\right)\leq {\frac {f(x)+f(y)}{2}}}$

and showed, with express reference to Cauchy's proof of the inequality of the arithmetic and geometric mean with forward-backward induction , that from this the relationship

${\displaystyle f\left({\frac {\sum _{i=1}^{n}x_{i}}{n}}\right)\leq {\frac {\sum _{i=1}^{n}f\left(x_{i}\right)}{n}}}$

for any natural numbers follows. From this he then concluded that ${\displaystyle n\;}$

${\displaystyle f\left({\frac {\sum _{i=1}^{n}k_{i}x_{i}}{\sum _{i=1}^{n}k_{i}}}\right)\leq {\frac {\sum _{i=1}^{n}k_{i}f\left(x_{i}\right)}{\sum _{i=1}^{n}k_{i}}}}$

for natural numbers and thus ${\displaystyle k_{i}\;}$

${\displaystyle f\left(\sum _{i=1}^{n}\lambda _{i}x_{i}\right)\leq \sum _{i=1}^{n}\lambda _{i}f\left(x_{i}\right)}$

for any rational and, if it is continuous, also real numbers between 0 and 1 with . ${\displaystyle f\;}$ ${\displaystyle \lambda _{i}\;}$ ${\displaystyle \sum _{i=1}^{n}\lambda _{i}=1}$

variants

• As for concave function , the function is convex, the Jensen's inequality is true for functions concave in the opposite direction, d. i.e., for every concave function and for positive with : ${\displaystyle f\;}$ ${\displaystyle -f\;}$ ${\displaystyle f\;}$ ${\displaystyle \lambda _{i}\;}$ ${\displaystyle \sum _{i=1}^{n}\lambda _{i}=1}$

${\displaystyle f\left(\sum _{i=1}^{n}\lambda _{i}x_{i}\right)\geq \sum _{i=1}^{n}\lambda _{i}f(x_{i}).}$

• The continuous variation of Jensen's inequality for the image of convex function is ${\displaystyle y\colon [a,b]\to \mathbb {R} }$ ${\displaystyle f\;}$

${\displaystyle f\left({\frac {1}{b-a}}\int _{a}^{b}y(x)\ \mathrm {d} x\right)\leq {\frac {1}{b-a}}\int _{a}^{b}f\left(y(x)\right)\ \mathrm {d} x.}$

• The continuous and discrete variation is in the measure-theoretic variant summarize: Is measure space with and is a μ-integrable real-valued function, then for each in the image of convex function ${\displaystyle \left(\Omega ,A,\mu \right)}$ ${\displaystyle \mu (\Omega )=1\;}$ ${\displaystyle y\;}$ ${\displaystyle y\;}$ ${\displaystyle f\;}$

${\displaystyle f\left(\int _{\Omega }y\ \mathrm {d} \mu \right)\leq \int _{\Omega }f\circ y\ \mathrm {d} \mu .}$

• Jensen's inequality is e.g. B. applicable for expected values . If convex and an integrable random variable then applies ${\displaystyle f\;}$ ${\displaystyle X\;}$

${\displaystyle f(\operatorname {E} (X))\leq \operatorname {E} (f(X)).}$

Analogous statements apply to the conditional expected value .

Applications

Jensen's inequality can be used, for example, to prove the inequality of the arithmetic and geometric mean and the Ky-Fan inequality . The variant for expected values ​​is used in stochastics to estimate certain random variables.

reversal

The statement of the measure theoretic variant of those inequality can be reversed in the following sense:

Let be a real function such that for every bounded (Lebesgue) measurable function holds ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle y\colon [0,1]\to \mathbb {R} }$

${\displaystyle f\left(\int _{0}^{1}y\ \mathrm {d} x\right)\leq \int _{0}^{1}f\circ y\ \mathrm {d} x}$,

then is convex. ${\displaystyle f}$

Web links

Wikisource: Otto Hölder: About a mean value rate (1889)  - Sources and full texts

• German abstract of Jensen's work
• Eric W. Weisstein : Jensen's Inequality . In: MathWorld (English).

Individual evidence

1. ^ ^{A b} Johan Ludwig William Valdemar Jensen : Sur les fonctions convexes et les inégalités entre les valeurs moyennes . In: Acta Math. Band 30 , 1906, pp. 175-193 , doi : 10.1007 / BF02418571 .
2. ↑ ^{a b} Otto Hölder : About a mean value proposition . In: News from the Royal. Society of Sciences and the Georg August University in Göttingen . From 1889., No. 1-21 . Dieterichsche Verlag-Buchhandlung, Göttingen 1889, p. 38 ff . (in Wikisource [accessed March 24, 2012]).
3. ^ Walter Rudin : Real and Complex Analysis . 3. Edition. McGraw-Hill, New York 1987, pp. 74 (English).