Newton's inequalities

In mathematics , Newton's inequalities are inequalities named after Isaac Newton , the author of the Philosophiae Naturalis Principia Mathematica .

statement

Let be real numbers and be the -th elementary symmetric polynomials in , defined by ${\displaystyle a_{1},\dots ,a_{n}}$ ${\displaystyle \sigma _{k}}$ ${\displaystyle k}$ ${\displaystyle a_{1},\dots ,a_{n}}$

${\displaystyle \sigma _{k}=\sum _{S\subseteq \{1,\dotsc ,n\} \atop \#S=k}\ \prod _{i\in S}a_{i}\ .}$

For example, and . ${\displaystyle \sigma _{1}=a_{1}+\dotsb +a_{n}}$ ${\displaystyle \sigma _{n}=a_{1}\dotsm a_{n}}$

Then fulfill the elementary symmetrical means defined by

${\displaystyle S_{k}={\frac {\sigma _{k}}{\binom {n}{k}}},}$

the inequalities

${\displaystyle S_{k-1}S_{k+1}\leq S_{k}^{2}}$

for all integers . ${\displaystyle 0<k<n}$

If all are not equal to zero, then equality holds if and only if all are equal. It should be noted that the arithmetic mean and the th power of the geometric mean of is. ${\displaystyle a_{i}}$ ${\displaystyle a_{i}}$ ${\displaystyle S_{1}}$ ${\displaystyle S_{n}}$ ${\displaystyle n}$ ${\displaystyle a_{i}}$

See also

• Maclaurin's inequality

Individual evidence

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