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The finite form of the equation was the logo of Institute For Mathematical Sciences at University Of Copenhagen until 2006 .


STATEMENTS


The classical form of Jensen's inequality involves several numbers and weights. The inequality can be stated quite generally using Measure Theory , and can be further generalized to its ''full strength'' in a probabilistic setting.


Finite form

For a real Convex Function φ, numbers ''xi'' in its domain, and positive weights ''ai'', Jensen's inequality can be stated as:

:\phi\left( rac{\sum a_{i} x_{i}}{\sum a_{i}} ight) \le rac{\sum a_{i} \phi (x_{i})}{\sum a_{i}};
and the inequality is clearly reversed if φ is Concave .

As a particular case, if the weights ''ai'' are all equal to unity, then

:\phi\left( rac{\sum x_{i}}{n} ight) \le rac{\sum \phi (x_{i})}{n}.

For instance, the \log(x) function is ''concave'', so substituting \phi(x)=-\log(x) in the previous formula, this establishes the (logarithm of) the familiar Arithmetic Mean-geometric Mean Inequality :

: rac{x_1 + x_2 + \cdots + x_n}{n} \ge \sqrt {Link without Title} {x_1 x_2 \cdots x_n}.

The variable ''x'' may, if required, be a function of another variable (or set of variables) ''t'', so that x_i = g(t_i). All of this carries directly over to the general continuous case: the weights ''ai'' are replaced by a non-negative integrable function ''f''(''x''), such as a probability distribution, and the summations replaced by integrals.


In measure-theoretic notation

Let (Ω,A,μ) be a Measure Space , such that μ(Ω) = 1. If ''g'' is a Real -valued function that is μ- Integrable , and if φ is a Measurable Convex Function on the real axis, then:

: arphi\left(\int_{\Omega} g\, d\mu ight) \le \int_\Omega arphi \circ g\, d\mu.


In probability-theory notation (real space)

The same result can be stated in a Probability Theory setting. Let (\Omega, \mathfrak{F},\mathbb{P}) be a Probability Space , X an Integrable real-valued Random Variable and φ a measurable Convex Function . Then:

: arphi\left(\mathbb{E}\{X\} ight) \leq \mathbb{E}\{ arphi(X)\}.

In this probability setting, the measure μ is intended as a probability \mathbb{P}, the integral with respect to μ as an Expected Value \mathbb{E}, and the function ''g'' as a Random Variable X.


In probability-theory notation (general)


More generally, let ''T'' be a real \mathfrak{G} of \mathfrak{F}:

  Here <math>\mathbb{E}\{\cdot\mathfrak{G} \}</math> Stands For The "http://wwwinformationdelightinfo/information/entry/conditional_expectation" class="copylinks">Expectation Conditioned to the &sigma-algebra <math>\mathfrak{G}</math> This general statement reduces to the previous ones when the topological vector space ''T'' is the Real Axis , and <math>\mathfrak{G}</math> is the trivial &sigma-algebra <math>\{\emptyset, \Omega\}</math>
  In Particular, For An Arbitrary Sub-&sigma-algebra <math>\mathfrak{G}</math> We Can Evaluate The Last Inequality When <math>x \mathbb{E}\{X\mathfrak{G}\},\,y=X-\mathbb{E}\{X\mathfrak{G}\}</math> to obtain:
  :<math>\mathbb{E}\{\left "(D arphi)(\mathbb{E}\{X\mathfrak{G}\})\cdot" class="copylinks" target="_blank">(X-\mathbb{E}\{X\mathfrak{G}\}) ight \mathfrak{G}\}=(D arphi)(\mathbb{E}\{X\mathfrak{G}\})\cdot \mathbb{E}\{ \left( X-\mathbb{E}\{X\mathfrak{G}\} ight) \mathfrak{G}\}=0,</math>
  :<math>\delta {1}(X) \Bbb{E}_{ heta}\{\delta(X') \,\, T(X')= T(X)\},</math>