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Cauchy–Schwarz inequality. The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) [1][2][3][4] is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.
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.
Visual proof that (x + y)2 ≥ 4xy. Taking square roots and dividing by two gives the AM–GM inequality. [1] 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 ...
Cauchy's mean value theorem, also known as the extended mean value theorem, [6] is a generalization of the mean value theorem. It states: if the functions f {\displaystyle f} and g {\displaystyle g} are both continuous on the closed interval [ a , b ] {\displaystyle [a,b]} and differentiable on the open interval ( a , b ) {\displaystyle (a,b ...
In mathematics, the rearrangement inequality[1] states that for every choice of real numbers and every permutation of the numbers we have. . (1) Informally, this means that in these types of sums, the largest sum is achieved by pairing large values with large values, and the smallest sum is achieved by pairing small values with large values ...
In algebra, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy, states that [ 1 ] for every choice of real or complex numbers (or more generally, elements of a commutative ring). Setting ai = ci and bj = dj, it gives Lagrange's identity, which is a stronger version of the Cauchy–Schwarz inequality ...
The special case p = q = 2 gives a form of the Cauchy–Schwarz inequality. [1] Hölder's inequality holds even if ‖ fg ‖ 1 is infinite, the right-hand side also being infinite in that case. Conversely, if f is in L p (μ) and g is in L q (μ), then the pointwise product fg is in L 1 (μ).
Hellinger distance. In probability and statistics, the Hellinger distance (closely related to, although different from, the Bhattacharyya distance) is used to quantify the similarity between two probability distributions. It is a type of f -divergence. The Hellinger distance is defined in terms of the Hellinger integral, which was introduced by ...