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Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps [27]) — For a 2-positive map between C*-algebras, for all , in its domain, () ‖ ‖ (), ‖ ‖ ‖ ‖ ‖ ‖. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:
In some more difficult applications of the method, some ingenuity might be required in order to choose the random variables X n for which the argument can be carried through. The Paley–Zygmund inequality is sometimes used instead of the Cauchy–Schwarz inequality and may occasionally give more refined results.
Proof. First equation: ... be a random variable with probability density function (; ... The Cauchy–Schwarz inequality shows that ...
The sign of the covariance of two random variables X and Y. ... holds via the Cauchy–Schwarz inequality. Proof: If () =, then it holds trivially. Otherwise, let ...
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).
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.
In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997, [1] to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu ...
He is credited with an early discovery of the Cauchy–Schwarz inequality, proving it for the infinite dimensional case in 1859, many years prior to Hermann Schwarz's research on the subject. Bunyakovsky is an author of Foundations of the mathematical theory of probability (1846). [7] Bunyakovsky published around 150 research papers. [1]