Search results
Results from the WOW.Com Content Network
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, [ 1 ] building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. [ 2 ]
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which ...
There are three inequalities between means to prove. There are various methods to prove the inequalities, including mathematical induction, the Cauchy–Schwarz inequality, Lagrange multipliers, and Jensen's inequality. For several proofs that GM ≤ AM, see Inequality of arithmetic and geometric means.
The proof of this inequality follows from the above combined with Klein's inequality. ... satisfies Jensen's Operator Inequality if the following holds ...
The proof follows from Jensen's inequality, making use of the fact the logarithm is concave: = = = =. By applying the exponential function to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get ∏ i = 1 n x i w i ≤ ∑ i = 1 n w i x i . {\displaystyle \prod _{i=1}^{n}x ...
The finite form of Jensen's inequality is a special case of this result. Consider the real numbers x 1, …, x n ∈ I and let := + + + denote their arithmetic mean.Then (x 1, …, x n) majorizes the n-tuple (a, a, …, a), since the arithmetic mean of the i largest numbers of (x 1, …, x n) is at least as large as the arithmetic mean a of all the n numbers, for every i ∈ {1, …, n − 1}.
The main tool in the proof of the above equivalence is the following result. [2] The following statements are equivalent ω ∈ A p for some 1 ≤ p < ∞. There exist 0 < δ, γ < 1 such that for all balls B and subsets E ⊂ B, |E| ≤ γ |B| implies ω(E) ≤ δ ω(B). There exist 1 < q and c (both depending on ω) such that for all balls B ...
In information theory, Gibbs' inequality is a statement about the information entropy of a discrete probability distribution. Several other bounds on the entropy of probability distributions are derived from Gibbs' inequality, including Fano's inequality. It was first presented by J. Willard Gibbs in the 19th century.