Jensen's inequality

The Jensen's inequality is an elementary inequality for convex and concave functions. It is because of their generality basis of many important inequalities, especially in analysis and information theory. The inequality is named after the Danish mathematician Johan Ludwig Jensen, she presented at a conference of the Danish Mathematical Society on 17 January 1905. Under somewhat different circumstances she finds herself in 1889 with Otto Hölder.

The Jensen's inequality states that the function value of a convex function on 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 ​​of the N points is greater than or equal to the function value at the middle of the n points.

Set

For a convex function and for non-negative with the following applies:

Proof by induction

If one uses the now common definition of " convex" that

Applies to all real between 0 and 1, then the Jensen inequality yields simply by induction on the number of support points.

Proof of Hölder

Hölder not yet used the term convex and pointed that out or monotonically increasing, the inequality

Follows for positive, in which he proved this mainly with the mean value theorem of differential calculus.

Proof of Jensen

Jensen went from the weaker definition

And showed with specific reference to the Cauchy's proof of the inequality of the arithmetic and geometric means with forward-backward induction that from the relationship

Follows for arbitrary natural numbers. From this he then further reasoned that

For natural numbers and thus

For any rational and, if is continuous, and real numbers between 0 and 1 shall apply with.

Variants

• Since the function is convex to concave features, concave features applies to the Jensen 's inequality in the reverse direction, i.e., is valid for each concave function, and to positive, comprising:

• The continuous variation of jens between inequality for a convex function of the image is

• The continuous and discrete variation is in the measure theoretic variant summarize: There is a measure space and μ is a - integrable real-valued function, then for every convex function in the image of the inequality

• The Jensen's inequality, for example, applicable to expected values ​​. Is convex and an integrable random variable, then applies

Applications

The Jensen's inequality can be used for example to prove the inequality of the arithmetic and geometric means and the Ky Fan inequality.