Jensen's inequality

The classical form of Jensen's inequality involves several numbers and weights.

For the proof of the more general version, Jensen's inequality cannot be dispensed with.

The finite form of Jensen's inequality is a special case of this result.

The final step is to use Jensen's inequality to move the expectation inside the function:

Jensen's inequality can also be proven graphically, as illustrated on the third diagram.

In particular cases inequalities can be proven also by Jensen's inequality:

That the ratio is biased can be shown with Jensen's inequality as follows:

Jensen is mostly renowned for his famous inequality, Jensen's inequality.

Using this, and the Jensen's inequality we get:

The operator version of Jensen's inequality is due to C. Davis.

In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.

Then Jensen's inequality becomes the following statement about convex integrals:

It generalizes the discrete form of Jensen's inequality.

A convex function of a martingale is a submartingale, by Jensen's inequality.

The result can alternatively be proved using Jensen's inequality or log sum inequality.

This equivalence can be verified by using Jensen's Inequality.

This bound was proved by Mallows, who used Jensen's inequality twice, as follows.

Jensen's inequality can be proved in several ways, and three different proofs corresponding to the different statements above will be offered.

The average absolute deviation from the mean is less than or equal to the standard deviation; one way of proving this relies on Jensen's inequality.

Jensen's inequality is of particular importance in statistical physics when the convex function is an exponential, giving:

The proof of Hoeffding's lemma uses Taylor's theorem and Jensen's inequality.

A notable inequality concerning this topic is Jensen's inequality, involving expected values of convex (or concave) functions.

The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex.

It is similar to Jensen's inequality and was found in 1965 by Tiberiu Popoviciu, a Romanian mathematician.

This equivalence can be established by using the exponential characterization of weights above, Jensen's inequality, and the John-Nirenberg inequality.