# Newton's and Maclaurin's Inequalities

### Symmetric functions and averages

Symmetric (or elementary symmetric) polynomials arise in Viète's formulas for the roots of polynomial equations. The formulas relate the coefficients of the polynomial to its roots. An $n\text{-th}$ degree polynomial has $n$ roots and $n+1$ coefficients. A polynomial is said to be *monic* when its leading coefficient equals $1.$

For a set $X_n=\{x_1,x_2,\ldots,x_n\}$ of $n$ variables, denote by $X_{n}^{k}$ the set of all **products** of the $n$ variables taken $k$ at a time. There are $n\choose k$ such products: $\left|X_{n}^{k}\right|={n\choose k}.$

$\displaystyle\begin{align} e_0(x_1,x_2,\ldots ,x_n) &= 1,\\ e_1(x_1,x_2,\ldots ,x_n) &= \sum_{t\in X_{n}^{1}}t = x_1+x_2+\ldots+x_n,\\ e_2(x_1,x_2,\ldots ,x_n) &= \sum_{t\in X_{n}^{2}}t = x_1x_2+x_1x_3+\ldots+x_{n-1}x_n,\\ &\cdots \cdots\\ e_n(x_1,x_2,\ldots ,x_n) &= \sum_{t\in X_{n}^{n}}t = x_1x_2\cdots x_n, \end{align}$

and their averages as

$\displaystyle\begin{align} E_0(X) &= E_0(x_1,x_2,\ldots ,x_n) = \frac{1}{{n\choose 0}}e_0(x_1,x_2,\ldots ,x_n),\\ E_1(X) &= E_1(x_1,x_2,\ldots ,x_n) = \frac{1}{{n\choose 1}}e_1(x_1,x_2,\ldots ,x_n),\\ E_2(X) &= E_0(x_1,x_2,\ldots ,x_n) = \frac{1}{{n\choose 2}}e_0(x_1,x_2,\ldots ,x_n),\\ &\cdots \cdots\\ E_n(X) &= E_n(x_1,x_2,\ldots ,x_n) = \frac{1}{{n\choose n}}e_n(x_1,x_2,\ldots ,x_n). \end{align}$

Newton's and Maclaurin's inequalities deal with those averages, although they can be rewritten in terms of the symmetric functions themselves.

### Newton's and Maclaurin's Inequalities

To formulate both Newton's inequality and Maclaurin's inequality we assume that all $x_i,$ $i=1,\ldots,n$ are non-negative.

**Newton's inequality**

$\left(E_k(X)\right)^2\gt E_{k-1}(X)E_{k+1}(X),$ $1\le k\le n-1.$

**Maclaurin's inequality**

$E_1(X)\gt \left(E_{2}(X)\right)^\frac{1}{2}\gt \left(E_{3}(X)\right)^\frac{1}{3}\gt\ldots\gt \left(E_{n}(X)\right)^\frac{1}{n}.$

The inequalities become equalities only when all $x_i$ are equal.

Note that $E_1(X)\gt \left(E_{n}(X)\right)^\frac{1}{n}$ is none other than the better known AM-GM inequality.

### Newton's inequality implies that of Maclaurin

Indeed, multiply Newton's inequalities in a sequence, as below:

$\displaystyle (E_0E_2)(E_1E_3)^2(E_2E_3)^3\cdots (E_{k-1}E_{k+1})^k\lt (E_1)^2(E_2)^4\cdots (E_k)^{2k}$

which reduces to $(E_k)^{k+1}\gt (E_{k+1})^k,$ or $\displaystyle (E_k)^{\frac{1}{k}}\gt (E_{k+1})^\frac{1}{k+1}.$

### Preliminaries for a proof of Newton's inequality

We'll prove a somewhat more general statement:

Suppose that $\alpha,\beta\gt 0$ are real and $j,k\ge 0$ integers such that

$\alpha+\beta=1$ and $j\alpha+k\beta\in\{0,1,\ldots,n\}.$

Then

$E_{j\alpha+k\beta}(X)\ge(E_j(X))^{\alpha}(E_k(X))^{\beta},$

for every $n$-tuple $X$ of non-negative real numbers. Moreover, equality occurs if and only if all $x_i$ are equal.

The proof is by induction on the number of variables $n.$

According to Rolle's theorem, if all roots of a polynomial $P(x)$ are real, or real and distinct, then the same is true for its derivative $P'(x).$ This remark will be applied to the polynomial

$\displaystyle P_X(x)=\prod_{i=1}^{n}(x-x_i)=\sum_{k=0}^{n}(-1)^{k}e_{k}(x_1,x_2,\ldots,x_n)x^{n-k}.$

If $y_1,y_2,\ldots,y_{n-1}$ the roots of the derivative of $P_X(x)$ then the $(n-1)-\text{tuple}$ $X'=\{y_1,y_2,\ldots,y_{n-1}\}$ is denoted $X'$ and is said to be derived from the $n-\text{tuple}$ $X.$

Observe that

$\displaystyle\prod_{i=1}^{n-1}(x-y_i)=\sum_{k=0}^{n_1}(-1)^ke_{k}(y_1,\ldots,y_{n-1})x^{n-1-k}$

and also that

$\displaystyle\begin{align} \prod_{i=1}^{n-1}(x-y_i) &=\frac{1}{n}\frac{P_X(x)}{dx}\\ &= \sum_{k=0}^{n_1}(-1)^k\frac{n-k}{n}e_{k}(x_1,\ldots,x_{n})x^{n-1-k}. \end{align}$

By comparing the two expressions we conclude that $\displaystyle e_{k}(y_1,\ldots,y_{n-1})=\frac{n-k}{n}e_{k}(x1,\ldots,x_{n}).$ Which gives the following

**Lemma 1**

$E_k(X)=E_k(X'),$ for every $k\in\{0,1,\ldots,|X|-1\}.$

Another simple and useful result is the following

**Lemma 2**

Assume that $X$ is an $n-\text{tuple}$ of real numbers that does not contain $0.$ Define $\displaystyle X^{-1}=\{\frac{1}{a}:\;a\in X\}.$ Then

$\displaystyle E_k(X^{-1})=\frac{E_{n-k}(X)}{E_n(X)},$

for every $k\in\{0,1,\ldots,n\}.$

### The induction

Now, for $|X|=2,$ we have to prove just one inequality:

$\displaystyle x_1x_2\le\left(\frac{x_1+x_2}{2}\right)^2,$

which the *AM-GM inequality* for $n=2,$ with the equality only when $x_1=x_2.$

Assume that the assertion holds for all $k-\text{tuples},$ with $k\le n-1.$ Let $|X|=n\ge 3,$ all $x_i$ non-negative and let $j,k$ non-negative integers, while $\alpha,\beta$ positive real numbers such that

$\alpha+\beta=1$ and $j\alpha+k\beta\in\{0,1,\ldots,n\}.$

According to Lemma 1,

$E_{j\alpha+k\beta}(X)\ge\left(E_j(X)\right)^{\alpha}\cdot\left(E_k(X)\right)^{\beta},$

except possibly for the case where $j\lt k=n,$ or $k\lt j=n.$ Assume, for example, that $j\lt k=n;$ then necessarily $j\alpha+n\beta\lt n.$ We have to show that

$E_{j\alpha+n\beta}(X)\ge\left(E_j(X)\right)^{\alpha}\cdot\left(E_n(X)\right)^{\beta}.$

If $0\in X,$ then $E_n(X)=0,$ and the inequality is obvious. The equality occurs only if $E_{j\alpha+n\beta}(X')=E_{j\alpha+n\beta}(X)=0,$ i.e. (according to the induction hypothesis), when all entries of $X$ coincide.

If none of $x_i$ is $0,$ then, by Lemma 2, we have to prove that

$E_{n-j\alpha-n\beta}(X^{-1})\ge\left(E_{n-j}(X^{-1})\right)^{\alpha},$

or equivalently (due to Lemma 1),

$E_{n-j\alpha-n\beta}((X^{-1})')\ge\left(E_{n-j}((X^{-1})')\right)^{\alpha}.$

The latter is true by virtue of our induction hypothesis.

### References

Constantin P. Niculescu, A NEW LOOK AT NEWTON’S INEQUALITIES, Journal of Inequalities in Pure and Applied Mathematics, Volume 1, Issue 2, Article 17, 2000

- An Inequality for Grade 8
- An Extension of the AM-GM Inequality
- Schur's Inequality
- Newton's and Maclaurin's Inequalities
- Rearrangement Inequality
- Chebyshev Inequality
- Jensen's Inequality
- Muirhead's Inequality
- Bergström's inequality
- Radon's Inequality and Applications
- Jordan and Kober Inequalities, PWW
- A Mathematical Rabbit out of an Algebraic Hat
- An Inequality With an Infinite Series
- An Inequality: 1/2 * 3/4 * 5/6 * ... * 99/100 less than 1/10
- A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
- An Inequality: Easier to prove a subtler inequality
- Inequality with Logarithms
- An inequality: 1 + 1/4 + 1/9 + ... less than 2
- Inequality with Harmonic Differences
- An Inequality by Uncommon Induction
- Hlawka's Inequality
- An Inequality in Determinants
- Application of Cauchy-Schwarz Inequality
- An Inequality from Tibet
- An Inequality with Constraint
- An Inequality from Morocco
- An Inequality for Mixed Means
- An Inequality in Integers
- An Inequality in Integers II
- An Inequality in Integers III
- An Inequality with Exponents
- Exponential Inequalities for Means
- A Simple Inequality in Three Variables
- An Asymmetric Inequality
- Linear Algebra Tools for Proving Inequalities
- An Inequality with a Generic Proof
- A Generalization of an Inequality from a Romanian Olympiad
- Area Inequality in Trapezoid
- Improving an Inequality
- RomanoNorwegian Inequality
- Inequality with Nested Radicals II
- Inequality with Powers And Radicals
- Inequality with Two Minima
- Simple Inequality with Many Faces And Variables
- An Inequality with Determinants
- An Inequality with Determinants II
- An Inequality with Determinants III
- An Inequality with Determinants IV
- An Inequality with Determinants V
- An Inequality with Determinants VI
- An Inequality with Determinants VII
- An Inequality in Reciprocals
- An Inequality in Reciprocals II
- An Inequality in Reciprocals III
- Monthly Problem 11199
- A Problem from the Danubius Contest 2016
- A Problem from the Danubius-XI Contest
- An Inequality with Integrals and Rearrangement
- An Inequality with Cot, Cos, and Sin
- A Trigonometric Inequality from the RMM
- An Inequality with Finite Sums
- Hung Viet's Inequality
- Hung Viet's Inequality II
- Hung Viet's Inequality III
- Inequality by Calculus
- Dorin Marghidanu's Calculus Lemma
- An Area Inequality
- A 4-variable Inequality from the RMM
- An Inequality from RMM with Powers of 2
- A Cycling Inequality with Integrals
- A Cycling Inequality with Integrals II
- An Inequality with Absolute Values
- An Inequality from RMM with a Generic 5
- An Elementary Inequality by Non-elementary Means
- Inequality in Quadrilateral
- Marian Dinca's Refinement of Nesbitt's Inequality
- An Inequality in Cyclic Quadrilateral
- An Inequality in Cyclic Quadrilateral II
- An Inequality in Cyclic Quadrilateral III
- An Inequality in Cyclic Quadrilateral IV
- Inequality with Three Linear Constraints
- Inequality with Three Numbers, Not All Zero
- An Easy Inequality with Three Integrals
- Divide And Conquer in Cyclic Sums
- Wu's Inequality
- A Cyclic Inequality in Three Variables
- Dorin Marghidanu's Inequality in Complex Plane
- Dorin Marghidanu's Inequality in Integer Variables
- Dorin Marghidanu's Inequality in Many Variables
- Dorin Marghidanu's Inequality in Many Variables Plus Two More
- Dorin Marghidanu's Inequality with Radicals
- Dorin Marghidanu's Light Elegance in Four Variables
- Dorin Marghidanu's Spanish Problem
- Two-Sided Inequality - One Provenance
- An Inequality with Factorial
- Wonderful Inequality on Unit Circle
- Quadratic Function for Solving Inequalities
- An Inequality Where One Term Is More Equal Than Others
- An Inequality and Its Modifications
- Complicated Constraint - Simple Inequality
- Distance Inequality
- Two Products: Constraint and Inequality
- The power of substitution II: proving an inequality with three variables
- Algebraic-Geometric Inequality
- One Inequality - Two Domains
- Radicals, Radicals, And More Radicals in an Inequality
- An Inequality in Triangle and In General
- Cyclic Inequality with Square Roots
- Dan Sitaru's Cyclic Inequality In Many Variables
- An Inequality on Circumscribed Quadrilateral
- An Inequality with Fractions
- An Inequality with Complex Numbers of Unit Length
- An Inequality with Complex Numbers of Unit Length II
- Le Khanh Sy's Problem
- An Inequality Not in Triangle
- An Acyclic Inequality in Three Variables
- An Inequality with Areas, Norms, and Complex Numbers
- Darij Grinberg's Inequality In Three Variables
- Small Change Makes Big Difference
- Inequality with Two Variables? Think Again
- A Problem From a Mongolian Olympiad for Grade 11
- Sitaru--Schweitzer Inequality
- An Inequality with Cyclic Sums And Products
- Problem 1 From the 2016 Pan-African Math Olympiad
- An Inequality with Integrals and Radicals
- Twin Inequalities in Four Variables: Twin 1
- Twin Inequalities in Four Variables: Twin 2
- Simple Inequality with a Variety of Solutions
- A Partly Cyclic Inequality in Four Variables
- Dan Sitaru's Inequality by Induction
- An Inequality in Three (Or Is It Two) Variables
- An Inequality in Four Weighted Variables
- An Inequality in Fractions with Absolute Values
- Inequalities with Double And Triple Integrals
- An Old Inequality
- Dan Sitaru's Amazing, Never Ending Inequality
- Leo Giugiuc's Exercise
- Another Inequality with Logarithms, But Not Really
- A Cyclic Inequality of Degree Four
- An Inequality Solved by Changing Appearances
- Distances to Three Points on a Circle
- An Inequality with Powers And Logarithm
- Four Integrals in One Inequality
- Same Integral, Three Intervals
- Dorin Marghidanu's Inequality with Generalization
- Dan Sitaru's Inequality with Three Related Integrals and Derivatives
- An Inequality in Two Or More Variables
- An Inequality in Two Or More Variables II
- A Not Quite Cyclic Inequality
- Dan Sitaru's Inequality: From Three Variables to Many in Two Ways
- An Inequality with Sines But Not in a Triangle
- An Inequality with Angles and Integers
- Sladjan Stankovik's Inequality In Four Variables
- An Inequality with Two Pairs of Triplets
- A Refinement of Turkevich's Inequality
- Dan Sitaru's Exercise with Pi and Ln
- Problem 4165 from Crux Mathematicorum
- Leo Giugiuc's Cyclic Quickie in Four Variables
- Dan Sitaru's Cyclic Inequality in Four Variables
- A Not Quite Cyclic Inequality from Tibet
- Three Variables, Three Constraints, Two Inequalities (Only One to Prove) - by Leo Giugiuc
- An inequality in 2+2 variables from SSMA magazine

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