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In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex -valued, continuous function on the contour Γ and if its absolute value | f ( z ) | is bounded by a constant M for all z on Γ , then
The second inequality is the elementary inequality between and . The last inequality follows by applying reverse Fatou lemma , i.e. applying the Fatou lemma to the non-negative functions g − f n {\displaystyle g-f_{n}} , and again (up to sign) cancelling the finite ∫ X g d μ {\displaystyle \int _{X}g\,d\mu } term.
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).
The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
Eaton's inequality, a bound on the largest absolute value of a linear combination of bounded random variables; Emery's inequality; Entropy power inequality; Etemadi's inequality; Fannes–Audenaert inequality; Fano's inequality; Fefferman's inequality; Fréchet inequalities; Gauss's inequality
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
Then | | + + + + + | | so | | + + + + + | | This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function is Lebesgue integrable only if all the four integrals are finite. So having a finite integral of the absolute value is equivalent to the conditions for ...
A power inequality is an inequality containing terms of the form a b, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises. Examples: