Newton's inequalities

Newton's inequalities

In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1a2, ..., an are real numbers and let σk denote the kth elementary symmetric function in a1a2, ..., an. Then the elementary symmetric means, given by

S_k = \frac{\sigma_k}{\binom{n}{k}}

satisfy the inequality

S_{k-1}S_{k+1}\le S_k^2

with equality if and only if all the numbers ai are equal. Note that S1 is simply the arithmetic mean, and Sn is of course the geometric mean.

See also

References

  • Newton, Isaac (1707). Arithmetica universalis: sive de compositione et resolutione arithmetica liber. 
  • D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55
  • Maclaurin, C. (1729). "A second letter to Martin Folks, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra,". Phil. Transactions, 36 (407–416): 59–96. doi:10.1098/rstl.1729.0011. 
  • Whiteley, J.N. (1969). "On Newton's Inequality for Real Polynomials". The American Mathematical Monthly (The American Mathematical Monthly, Vol. 76, No. 8) 76 (8): 905–909. doi:10.2307/2317943. JSTOR 2317943. 
  • Niculescu, Constantin (2000). "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics 1 (2). http://jipam.vu.edu.au/. 

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • List of inequalities — This page lists Wikipedia articles about named mathematical inequalities. Inequalities in pure mathematics =Analysis= * Askey–Gasper inequality * Bernoulli s inequality * Bernstein s inequality (mathematical analysis) * Bessel s inequality *… …   Wikipedia

  • Isaac Newton — Sir Isaac Newton …   Wikipedia

  • Elements of the Philosophy of Newton — is a book written by the philosopher Voltaire in 1738 that helped to popularize the theories and thought of Isaac Newton. This book, coupled with Letters on the English, written in 1733, demonstrated Voltaire had moved beyond the simple poetry… …   Wikipedia

  • MacLaurin's inequality — In mathematics, MacLaurin s inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a 1, a 2, ..., a n be positive real numbers, and for k = 1, 2, ..., n define the averages S k as follows …   Wikipedia

  • List of mathematics articles (N) — NOTOC N N body problem N category N category number N connected space N dimensional sequential move puzzles N dimensional space N huge cardinal N jet N Mahlo cardinal N monoid N player game N set N skeleton N sphere N! conjecture Nabla symbol… …   Wikipedia

  • Olga Holtz — bei einem Vortrag am Institute for Advanced Study in Princeton am 26. Oktober 2009 Olga Holtz (russisch Ольга Гольц; * 19. August 1973 in der Oblast Tscheljabinsk im Föderationskreis Ural) ist eine …   Deutsch Wikipedia

  • Integral — This article is about the concept of integrals in calculus. For the set of numbers, see integer. For other uses, see Integral (disambiguation). A definite integral of a function can be represented as the signed area of the region bounded by its… …   Wikipedia

  • Pierre-Simon Laplace — Laplace redirects here. For the city in Louisiana, see LaPlace, Louisiana. For the joint NASA ESA space mission, see Europa Jupiter System Mission. Pierre Simon, marquis de Laplace Pierre Simon Laplace (1749–1827). Posthumous portrait …   Wikipedia

  • Mathematical optimization — For other uses, see Optimization (disambiguation). The maximum of a paraboloid (red dot) In mathematics, computational science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to… …   Wikipedia

  • Linear programming — (LP, or linear optimization) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”