Hölder's Inequality | Brilliant Math & Science Wiki
Hölder's inequality is often used to deal with square (or higher-power) roots of expressions in inequalities since those can be eliminated through successive multiplication. Here is an example: Let a,b,c a,b,c be positive reals satisfying a+b+c=3 …
Classical Inequalities | Brilliant Math & Science Wiki
AM-GM Inequality. Main Article: AM-GM Inequality. The AM-GM inequality relates the arithmetic mean (AM) to the geometric mean (GM). The AM-GM inequality is commonly used in competition math to find the maximum or minimum value of a multi-variable function or expression. Given non-negative real numbers a_1,a_2,\cdots,a_n, a1,a2,⋯,an, the ...
Young's Inequality | Brilliant Math & Science Wiki
Young's inequality is a special case of the weighted AM-GM inequality. It is very useful in real analysis, including as a tool to prove Hölder's inequality. It is also a special case of a more general inequality known as Young's inequality for increasing functions. Click here for a proof As mentioned in the introduction, Young's inequality is essential in the proof of ...
Minkowski Dimension | Brilliant Math & Science Wiki
In the study of fractals, Minkowski dimension (a.k.a. box-counting dimension) is a notion of dimension for fractals, measuring how complexity of detail changes with the scale at which one views the fractal. Very roughly, the larger a fractal's Minkowski dimension, the more self-similar the fractal is, in the sense that the fractal contains many constricted copies of itself. …
Titu's Lemma | Brilliant Math & Science Wiki
It is a direct consequence of Cauchy-Schwarz inequality. This form is especially helpful when the inequality involves fractions where the numerator is a perfect square. It is obtained by applying the substitution \(a_i= \frac{x_i}{ \sqrt{y_i} }\) and \( b_i = \sqrt{y_i} \) into the Cauchy-Schwarz inequality. Equality holds if and only if \( a_i ...
Jensen's Inequality | Brilliant Math & Science Wiki
Jensen's Inequality. Jensen's inequality is an inequality involving convexity of a function. We first make the following definitions: I) I) lies above the graph. An example of a convex function is. f (x)=x^2 f (x) = x2. I) I) lies below the graph. An example of a …
Definite Integrals | Brilliant Math & Science Wiki
Definite Integrals. The red area is above the axis and is positive. The blue area is below the axis and is negative. A definite integral is a formal calculation of area beneath a function, using infinitesimal slivers or stripes of the region. Integrals may represent the (signed) area of a region, the accumulated value of a function changing ...
Arithmetic Mean - Geometric Mean | Brilliant Math & Science Wiki
Guided training for mathematical problem solving at the level of the AMC 10 and 12. The arithmetic mean-geometric mean (AM-GM) inequality states that the arithmetic mean of non-negative real numbers is greater than or equal to the geometric mean of the same list. Further, equality holds if and only if every number in the list is the same.
Hilbert Space | Brilliant Math & Science Wiki
Hilbert Space. A Hilbert space is a vector space V V equipped with an inner product, which can be thought of as a generalization of the dot product in Euclidean space, with the additional property that the metric coming from the inner product makes V V into a complete metric space. The basic example of a Hilbert space is {\mathbb R}^n Rn \big ...
Logarithmic Inequalities | Brilliant Math & Science Wiki
Logarithmic inequalities are inequalities in which one (or both) sides involve a logarithm. Like exponential inequalities, they are useful in analyzing situations involving repeated multiplication, such as in the cases of interest and exponential decay. The key to working with logarithmic inequalities is the following fact: If ...