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In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a1, a2, ..., an be positive real numbers, and for k = 1, 2, ..., n define the averages Sk as follows: The numerator of this fraction is the elementary symmetric polynomial of degree k in the n variables a1, a2, ..., an, that is, the sum of all products of k of the numbers a1, a2, ..., an with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient with equality if and only if all the ai are equal.

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  • Maclaurin-Ungleichung
  • Disuguaglianza di MacLaurin
  • Maclaurin's inequality
  • Inégalité de Maclaurin
  • 麦克劳林不等式
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  • Die Maclaurin-Ungleichung (nach Colin Maclaurin) ist eine Aussage aus der Analysis, einem Teilgebiet der Mathematik. Sie verschärft die Ungleichung vom arithmetischen und geometrischen Mittel, die besagt, dass das arithmetische Mittel von endlich vielen positiven reellen Zahlen stets mindestens so groß ist wie ihr geometrisches Mittel, in Formeln für eine natürliche Zahl und . In der Verschärfung werden noch weitere Mittelwerte angegeben, die zwischen dem arithmetischen und geometrischen Mittel liegen, beispielsweise besagt die Ungleichung für drei Zahlen
  • In matematica, la disuguaglianza di MacLaurin fornisce una serie di termini intermedi tra la media aritmetica e quella geometrica di una n-upla di reali positivi.
  • En mathématiques, l'inégalité de Maclaurin est une généralisation de l'inégalité arithmético-géométrique.
  • 数学中,麦克劳林不等式(Maclaurin's inequality),以科林·麦克劳林冠名,是算术几何平均不等式的加细。 设 a1, a2, ..., an 是正实数,对 k = 1, 2, ..., n 定义平均 Sk 为 这个分式的分子是度数为 n 变元 a1, a2, ..., an 的 k 阶基本对称多项式,即 a1, a2, ..., an 中指标递增的任意 k 个数乘积之和。分母是分子的项数,二项式系数 。 麦克劳林不等式是如下不等式链: 等号成立当且仅当所有 ai 相等。 对 n = 2,这个给出两个数通常的几何算术平均不等式。n = 4 的情形很好地展示了麦克劳林不等式:
  • In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a1, a2, ..., an be positive real numbers, and for k = 1, 2, ..., n define the averages Sk as follows: The numerator of this fraction is the elementary symmetric polynomial of degree k in the n variables a1, a2, ..., an, that is, the sum of all products of k of the numbers a1, a2, ..., an with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient with equality if and only if all the ai are equal.
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  • Die Maclaurin-Ungleichung (nach Colin Maclaurin) ist eine Aussage aus der Analysis, einem Teilgebiet der Mathematik. Sie verschärft die Ungleichung vom arithmetischen und geometrischen Mittel, die besagt, dass das arithmetische Mittel von endlich vielen positiven reellen Zahlen stets mindestens so groß ist wie ihr geometrisches Mittel, in Formeln für eine natürliche Zahl und . In der Verschärfung werden noch weitere Mittelwerte angegeben, die zwischen dem arithmetischen und geometrischen Mittel liegen, beispielsweise besagt die Ungleichung für drei Zahlen
  • In matematica, la disuguaglianza di MacLaurin fornisce una serie di termini intermedi tra la media aritmetica e quella geometrica di una n-upla di reali positivi.
  • En mathématiques, l'inégalité de Maclaurin est une généralisation de l'inégalité arithmético-géométrique.
  • In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a1, a2, ..., an be positive real numbers, and for k = 1, 2, ..., n define the averages Sk as follows: The numerator of this fraction is the elementary symmetric polynomial of degree k in the n variables a1, a2, ..., an, that is, the sum of all products of k of the numbers a1, a2, ..., an with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient Maclaurin's inequality is the following chain of inequalities: with equality if and only if all the ai are equal. For n = 2, this gives the usual inequality of arithmetic and geometric means of two numbers. Maclaurin's inequality is well illustrated by the case n = 4: Maclaurin's inequality can be proved using the Newton's inequalities.
  • 数学中,麦克劳林不等式(Maclaurin's inequality),以科林·麦克劳林冠名,是算术几何平均不等式的加细。 设 a1, a2, ..., an 是正实数,对 k = 1, 2, ..., n 定义平均 Sk 为 这个分式的分子是度数为 n 变元 a1, a2, ..., an 的 k 阶基本对称多项式,即 a1, a2, ..., an 中指标递增的任意 k 个数乘积之和。分母是分子的项数,二项式系数 。 麦克劳林不等式是如下不等式链: 等号成立当且仅当所有 ai 相等。 对 n = 2,这个给出两个数通常的几何算术平均不等式。n = 4 的情形很好地展示了麦克劳林不等式:
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  • MacLaurin's Inequality
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