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In mathematics, the QM-AM-GM-HM inequalities, also known as the mean inequality chain, state the relationship between the harmonic mean, geometric mean, arithmetic mean, and quadratic mean (also known as root mean square). Suppose that ,, …, are positive real numbers. Then
The harmonic mean is denoted by H in purple, while the arithmetic mean is A in red and the geometric mean is G in blue. Q denotes a fourth mean, the quadratic mean . Since a hypotenuse is always longer than a leg of a right triangle , the diagram shows that H ≤ G ≤ A ≤ Q {\displaystyle H\leq G\leq A\leq Q} .
In mathematics, the arithmetic–geometric mean (AGM or agM [1]) of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some ...
The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
A geometric construction of the quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by H, geometric by G, arithmetic by A and quadratic mean (also known as root mean square) denoted by Q. Comparison of the arithmetic, geometric and harmonic means of a pair of numbers.
In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we form the geometric mean of g 0 = x and h 0 = y and call it g 1, i.e. g 1 is the square root of xy. We also form the harmonic mean of x and y and call it h 1, i.e. h 1 is the reciprocal of the arithmetic mean of the reciprocals of ...
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians [4] because of their importance in geometry and music.
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) [1] are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means ( arithmetic , geometric , and harmonic means ).