QM-AM-GM-HM Inequalities

picture info	QM-AM-GM-HM Inequalities In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean (HM), geometric mean (GM), arithmetic mean (AM), and Root mean square, quadratic mean (QM; also known as root mean square). Suppose that x_1, x_2, \ldots, x_n are positive real numbers. Then : 0 0 implies the inequality: : \frac \leq \sqrt[n]. The ''n'' = 2 case When ''n'' = 2, the inequalities become :\frac \leq \sqrt \leq \frac\leq\sqrt for all x_1, x_2 > 0, which can be visualized in a semi-circle whose diameter is [''AB''] and center ''D''. Suppose ''AC'' = ''x''1 and ''BC'' = ''x''2. Construct perpendiculars to [''AB''] at ''D'' and ''C'' respectively. Join [''CE''] and [''DF''] and further construct a perpendicular [''CG''] to [''DF''] at ''G''. Then the length of ''GF'' can be calculated to be the harmonic mean, ''CF'' to be the geometric mean, ''DE'' to be the arithmetic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Inequality Of Arithmetic And Geometric Means Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups Income inequality, an unequal distribution of income Wealth inequality, an unequal distribution of wealth Spatial inequality, the unequal distribution of income and resources across geographical regions International inequality, economic differences between countries * Social inequality, unequal opportunities and rewards for different social positions or statuses within a group Gender inequality, unequal treatment or perceptions due to gender Racial inequality, social distinctions between racial and ethnic groups within a society * Health inequality, differences in the quality of health and healthcare across populations * Educational inequality, the unequal distribution of academic resources * Environmental inequality, unequal environmental harms between different neighbor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Inequalities (mathematics) Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups Income inequality, an unequal distribution of income Wealth inequality, an unequal distribution of wealth Spatial inequality, the unequal distribution of income and resources across geographical regions International inequality, economic differences between countries * Social inequality, unequal opportunities and rewards for different social positions or statuses within a group Gender inequality, unequal treatment or perceptions due to gender Racial inequality, social distinctions between racial and ethnic groups within a society * Health inequality, differences in the quality of health and healthcare across populations * Educational inequality, the unequal distribution of academic resources * Environmental inequality, unequal environmental harms between differe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Generalized Mean In mathematics, generalised means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic mean, arithmetic, geometric mean, geometric, and harmonic mean, harmonic means). Definition If is a non-zero real number, and x_1, \dots, x_n are positive real numbers, then the generalized mean or power mean with exponent of these positive real numbers is M_p(x_1,\dots,x_n) = \left( \frac \sum_^n x_i^p \right)^ . (See Norm (mathematics)#p-norm, -norm). For we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below): M_0(x_1, \dots, x_n) = \left(\prod_^n x_i\right)^ . Furthermore, for a sequence of positive weights we define the weighted power mean as M_p(x_1,\dots,x_n) = \left(\frac \right)^ and when , it is equal to the weighted geometric mean: M_0(x_1,\dots,x_n) = \left(\prod_^n x_i^\right)^ . The unweight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]